ELLIPTIC SPECTRA, THE WITTEN GENUS AND THE THEOREM OF THE CUBE
M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
Contents
1. Introduction *
* 2
1.1. Outline of the paper *
* 6
2. More detailed results *
* 6
2.1. The algebraic geometry of even periodic ring spectra *
* 6
2.2. Constructions of elliptic spectra *
* 8
2.3. The complex-orientable homology of BU<2k> for k 3 *
* 9
2.4. The complex-orientable homology of MU<2k> for k 3 *
* 13
2.5. The oe-orientation of an elliptic spectrum *
* 17
2.6. The Tate curve *
* 19
2.7. The elliptic spectrum KTateand its oe-orientation *
* 21
2.8. Modularity *
* 24
3. Calculation of C_k(bGa; Gm ) *
* 24
3.1. The cases k = 0 and k = 1 *
* 25
3.2. The strategy for k = 2 and k = 3 *
* 25
3.3. The case k = 2 *
* 26
3.4. The case k = 3: statement of results *
* 27
3.5. Additive cocycles *
* 29
3.6. Multiplicative cocycles *
* 31
3.7. The Weil pairing: cokernel of ffix :C_2(bGa; Gm ) !C_3(bGa; Gm ) *
* 33
3.8. The map ffix :C_1(bGa; Gm ) !C_2(bGa; Gm ) *
* 37
3.9. Rational multiplicative cocycles *
* 38
4. Topological calculations *
* 38
4.1. Ordinary cohomology *
* 39
4.2. The isomorphism for BU<0> and BU<2> *
* 39
4.3. The isomorphism for rational homology and all k *
* 40
4.4. The ordinary homology of BSU *
* 40
4.5. The ordinary homology of BU<6> *
* 41
4.6. BSU and BU<6> for general E *
* 44
Appendix A. Additive cocycles *
* 45
A.1. Rational additive cocycles *
* 45
A.2. Divisibility *
* 46
1
2 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
A.3. Additive cocycles: The modular case *
* 48
Appendix B. Generalized elliptic curves *
* 50
B.1. Examples of Weierstrass curves *
* 55
B.2. Elliptic curves over C *
* 57
B.3. Singularities *
* 58
B.4. The cubical structure for the line bundle I(0) on a generalized ellipti*
*c curve 58
References *
* 62
1. Introduction
This paper is part of a series ([HMM98 , HM98] and other work in progress) ge*
*tting at some new aspects
of the topological approach to elliptic genera. Most of these results were anno*
*unced in [Hop95].
In [Och87] Ochanine introduced the elliptic genus_a cobordism invariant of or*
*iented manifolds taking its
values in the ring of (level 2) modular forms. He conjectured and proved half o*
*f the rigidity theorem_that
the elliptic genus is multiplicative in bundles of spin manifolds with connecte*
*d structure group.
Ochanine defined his invariant strictly in terms of characteristic classes, a*
*nd the question of describing the
elliptic genus in more geometric terms naturally arose_especially in connection*
* with the rigidity theorem.
In [Wit87, Wit88] Witten interpreted Ochanine's invariant in terms of index t*
*heory on loop spaces and
offered a proof of the rigidity theorem. Witten's proof was made mathematicall*
*y rigorous by Bott and
Taubes [BT89], and since then there have been several new proofs of the rigidit*
*y theorem [Liu95, Ros98].
In the same papers Witten described a variant of the elliptic genus now known*
* as the Witten genus. There
is a characteristic class of Spin manifolds, twice which is the first Pontrjag*
*in class, p1. The Witten genus
is a cobordism invariant of Spin-manifolds for which = 0, and it takes its val*
*ues in modular forms (of level
1). It has exhibited a remarkably fecund relationship with geometry (see [Seg88*
*], and [HBJ92]).
Rich as it is, the theory of the Witten genus is not as developed as are the *
*invariants described by the
index theorem. One thing that is missing is an understanding of the Witten genu*
*s of a family. Let S be a
space, and Msa family of n-dimensional Spin-manifolds (with = 0) parameterized*
* by the points of S. The
family Ms defines an element in the cobordism group
MO<8>-nS;
where MO<8> denotes the cobordism theory of "Spin-manifolds with = 0." The Wi*
*tten genus of this
family should be some kind of "family of modular forms" parameterized by the po*
*ints of S. Motivated by
the index theorem, we should regard this family of modular forms as an element *
*in
E-nS
for some (generalized) cohomology theory E. From the topological point of view,*
* the Witten genus of a
family is thus a multiplicative map of generalized cohomology theories
MO<8> -!E;
and the question arises as to which E to choose, and how, in this language, to *
*express the modular invariance
of the Witten genus. One candidate for E, elliptic cohomology, was introduced b*
*y Landweber, Ravenel, and
Stong in [LRS95].
To keep the technicalities to a minimum, we focus in this paper on the restri*
*ction of the Witten genus to
stably almost complex manifolds with a trivialization of the Chern classes c1 a*
*nd c2 of the tangent bundle.
The bordism theory of such manifolds is denoted MU<6>. We will consider general*
*ized cohomology theories
(or, more precisely, homotopy commutative ring spectra) E which are even and pe*
*riodic. In the language of
generalized cohomology, this means that the cohomology groups
E"0(Sn)
ELLIPTIC SPECTRA *
* 3
are zero for n odd, and that for each pointed space X, the map
"E0(S2) "E0(X) -!"E0(S2^ X)
E0(pt)
is an isomorphism. In the language of spectra the conditions are that
ssoddE = 0
and that ss2E contains a unit. Our main result is a convenient description of a*
*ll multiplicative maps
MU<6> -!E:
In another paper in preparation we will give, under more restrictive hypotheses*
* on E, an analogous description
of the multiplicative maps
MO<8> -!E:
These results lead to a useful homotopy theoretic explanation of the Witten g*
*enus, and to an expression
of the modular invariance of the Witten genus of a family. To describe them it *
*is necessary to make use of
the language of formal groups.
The assumption that E is even and periodic implies that the cohomology ring
E0CP1 :
is the ring of functions on a formal group PE over ss0E = E0(pt) [Qui69, Ada74]*
*. From the point of view
of the formal group, the result [Ada74, Part II, Lemma 4.6] can be interpreted *
*as saying that the set of
multiplicative maps
MU -!E
is naturally in one to one correspondence with the set of rigid sections of a c*
*ertain rigid line bundle 1(L)
over PE. Here a line bundle is said to be rigid if it has a specified trivializ*
*ation at the zero element, and a
section is said to be rigid if it takes the specified value at zero. Our line b*
*undle L is the one whose sections
are functions that vanish at zero, or in other words L = O(-{0}). The fiber of *
*1(L) at a point a 2 PE is
defined to be L0 L*a; it is immediate that 1(L) has a canonical rigidification.
Similarly, given a line bundle L over a commutative group A, let 3(L) be the *
*line bundle over A3 whose
fiber at (a; b; c) is
3(L)(a;b;c)= La+bLb+cLa+cL0_L:
a+b+cLaLbLc
In this expression the symbol "+" refers to the group law of A, and multiplicat*
*ion and division indicate the
tensor product of lines and their duals. A cubical structure on L is a nowhere *
*vanishing section s of 3(L)
satisfying (after making the appropriate canonical identifications of line bund*
*les)
(rigid) s(0; 0; 0)=1
(symmetry) s(aoe(1); aoe(2);=aoe(3))s(a1; a2; a3)
(cocycle) s(b; c; d)s(a; b=+ c;sd)(a + b; c; d)s(a; b; d):
(See [Bre83], and Remark 2.42 for comparison of conventions.) Our main result (*
*2.50) asserts that the set
of multiplicative maps
MU<6> -!E
is naturally in one to one correspondence with the set of cubical structures on*
* L = O(-{0}).
We have chosen a computational approach to the proof of this theorem partly b*
*ecause it is elementary,
and partly because it leads to a general result. In [AS98], the first and third*
* authors give a less computational
proof of this result (for formal groups of finite height in positive characteri*
*stic), using ideas from [Mum65 ,
Gro72, Bre83] on the algebraic geometry of biextensions and cubical structures.
On an elliptic curve the line bundle O(-{0}) has a unique cubical structure. *
*Indeed, for fixed x and y,
there is by Abel's theorem a rational function f(x; y; z) with divisor {-x-y}+{*
*0}-{-x}-{-y}. Any two
such functions have a constant ratio, so the quotient s(x; y; z) = f(x; y; 0)=f*
*(x; y; z) is well-defined and is
easily seen to determine a trivialization of 3(O(-{0})). Since the only global *
*functions on an elliptic curve
4 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
are constants, the requirement s(0; 0; 0) = 1 determines the section uniquely, *
*and shows that it satisfies the
"symmetry" and "cocycle" conditions. In fact the "theorem of the cube" (see for*
* example [Mum70 ]) shows
more generally that any line bundle over any abelian variety has a unique cubic*
*al structure.
Over the complex numbers, a transcendental formula for f(x; y; z) is
oe(x_+_y_+_z)_oe(z);
oe(x + y) oe(x + z)
where oe is the Weierstrass oe function. It follows that the unique cubical str*
*ucture is given by
oe(x_+_y)_oe(x_+_z)_oe(y_+:z) oe(0) (1*
*.1)
oe(x + y + z) oe(x) oe(y) oe(z)
Putting all of this together, if the formal group PE can be identified with t*
*he formal completion of an
elliptic curve, then there is a canonical multiplicative map
MU<6> -!E
corresponding to the unique cubical structure which extends to the elliptic cur*
*ve.
Definition 1.2.An elliptic spectrum consists of
i.an even, periodic, homotopy commutative ring spectrum E with formal group P*
*E over ss0E;
ii.a generalized elliptic curve C over E0(pt);
iii.an isomorphism t:PE -!Cbof PE with the formal completion of C.
For an elliptic spectrum E = (E; C; t), the oe-orientation
oeE :MU<6> -!E
is the map corresponding to the unique cubical structure extending to C.
Note that this definition involves generalized elliptic curves over arbitrary*
* rings. The relevant theory is
developed in [KM85 , DR73]; we give a summary in Appendix B.
A map of elliptic spectra E1 = (E1; C1; t1) -! E2 = (E2; C2; t2) consists of *
*a map f :E1 -! E2 of
multiplicative cohomology theories, together with an isomorphism of elliptic cu*
*rves
(ss0f)*C2-! C1;
extending the induced map of formal groups. Given such a map, the uniqueness of*
* cubical structures over
elliptic curves shows that
MU<6>G (1*
*.3)
oeE1wwww GGoeE2GG
ww GGG
--www G##
E1 _______f_______//E2
commutes. We will refer to the commutativity of this diagram as the modular inv*
*ariance of the oe-orientation.
By way of illustration, let's consider examples derived from elliptic curves *
*over C, and ordinary cohomology
(for which the formal group is the additive group).
An elliptic curve over C is of the form C= for some lattice C. The map of f*
*ormal groups derived
from
C -!C=
gives an isomorphism t , from the additive formal group to the formal completio*
*n of the elliptic curve. Let
R be the graded ring C[u ; u-1] with |u | = 2, and define an elliptic spectrum*
* H = (E ; C ; tb)y taking
E to be the spectrum representing
H*( - ; R );
C the elliptic curve C=, and t the isomorphism described above.
ELLIPTIC SPECTRA *
* 5
The abelian group of cobordism classes of 2n-dimensional stably almost comple*
*x manifolds with a trivi-
alization of c1 and c2 is
MU<6>2n(pt):
The oe-orientation for H thus associates to each such M, an element of (E )2n(*
*pt) which can be written
(M; ) . un;
with
(M; ) 2 C:
Suppose that 0 C is another lattice, and that is a non-zero complex number f*
*or which . = 0.
Then multiplication by gives an isomorphism C= -!C=0. This extends to a map H0*
* -! H , of elliptic
spectra, which, in order to induce the correct map of formal groups, must send *
*u0 to u (this is explained
in example 2.3). The modular invariance of the oe-orientation then leads to the*
* equation
(M; . ) = -n(M; ):
This can be put in a more familiar form by choosing a basis for the lattice .*
* Given a complex number
o with positive imaginary part, let (o) be the lattice generated by 1 and o, an*
*d set
f(M; o) = (M; (o)):
Given
a b
c d 2 SL2(Z)
set
= (o)
0= ((a o + b)=(c o + d))
= (c o + d)-1:
The above equation then becomes
f(M; (a o + b)=(c o + d)) = (c o + d)nf(M; o);
which is the functional equation satisfied by a modular form of weight n. It ca*
*n be shown that f(M; o) is a
holomorphic function of o by considering the elliptic spectrum derived from the*
* family of elliptic curves
H x C=<1; o> -!H
parameterized by the points of the upper half plane H, and with underlying homo*
*logy theory
-1
H* - ; O[u; u ;]
where O is the ring of holomorphic functions on H. Thus the oe-orientation asso*
*ciates a modular form of
weight n to each 2n-dimensional MU<6>-manifold. Using an elliptic spectrum cons*
*tructed out of K-theory
and the Tate curve, one can also show that the modular forms that arise in this*
* manner have integral
q-expansions (see x2.8).
In fact, it follows from formula (1.1)(for details see x2.7) that the q-expan*
*sion of this modular form is
the Witten genus of M. The oe-orientation can therefore be viewed as a topologi*
*cal refinement of the Witten
genus, and its modular invariance (1.3), an expression of the modular invarianc*
*e of the Witten genus of a
family.
All of this makes it clear that one can deduce special properties of the Witt*
*en genus by taking special
choices of E. But it also suggests that the really natural thing to do is to co*
*nsider all elliptic curves at once.
This leads to some new torsion companions to the Witten genus, some new congrue*
*nces on the values of the
Witten genus, and to the ring of topological modular forms. It is the subject o*
*f the papers [HMM98 , HM98].
6 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
1.1. Outline of the paper. In x2, we state our results and the supporting defin*
*itions in more detail. In
x2.3 we give a detailed account of our algebraic model for E0BU<2k>. In x2.4 we*
* describe our algebraic
model for E0MU<2k>. We deduce our results about MU<2k> from the results about B*
*U<2k> and careful
interpretation of the Thom isomorphism; the proof of the main result about E0BU*
*<2k> (Theorem 2.29) is
the subject of x4.
In x2.5 we give in more detail the argument sketched in the introduction that*
* there is a unique cubical
structure on any elliptic curve. We give an argument with explicit formulae whi*
*ch works when the elliptic
curves in question are allowed to degenerate to singular cubics ("generalized e*
*lliptic curves"), and also gives
some extra insight even in the non-degenerate case. The proof of the main formu*
*la (Proposition 2.55) is
given in appendix B.
In x2.6, we give a formula for the cubical structure on the Tate curve, inspi*
*red by the transcendental
formula involving the oe-function that was mentioned in the introduction. In x2*
*.7, we interpret this formula
as describing the oe-orientation for the elliptic spectrum KTate, and we show t*
*hat its effect on homotopy rings
is the Witten genus. In x2.8, we deduce the modularity of the Witten genus from*
* the modular invariance of
the oe-orientation.
The rest of the main body of the paper assembles a proof of Theorem 2.29. In*
* x3 we study a set
C_k(bGa; Gm )(R) of formal power series in k variables over a ring R with certa*
*in symmetry and cocycle
properties. This is a representable functor of R, in other words C_k(bGa; Gm ) *
*is an affine group scheme. For
0 k 3 we will eventually identify C_k(bGa; Gm ) with spec(H*BU<2k>). For k = *
*3 we use a small fragment
of the theory of Weil pairings associated to cubical structures; this forms the*
* heart of an alternative proof
of our results [AS98] which works for p-divisible formal groups but not for the*
* formal group of an arbitrary
generalized elliptic curve.
In x4 we first check that our algebraic model coincides with the usual descri*
*ption of spec(E0BU). We
then compare our algebraic calculations to the homology of the fibration
BSU -!BU -!CP1
to show that spec(H*BSU) ~=C_2(bGa; Gm ).
We then recall Singer's analysis of the Serre spectral sequence of the fibrat*
*ion
K(Z; 3) -!BU<6> -!BSU:
By identifying the even homology of K(Z; 3) with the scheme of Weil pairings de*
*scribed in x3.7, we show
that spec(H*BU<6>) ~=C_3(bGa; Gm ). Finally we deduce Theorem 2.29 for all E fr*
*om the case of ordinary
homology.
The paper has two appendices. The first proves some results about the group *
*of additive cocycles
C_k(bGa; bGa)(A), which are used in x3. The second gives an exposition of the *
*theory of generalized ellip-
tic curves, culminating in a proof of Proposition 2.55. We have tried to make t*
*hings as explicit as possible
rather than relying on the machinery of algebraic geometry, and we have given a*
* number of examples.
2. More detailed results
2.1. The algebraic geometry of even periodic ring spectra. Let BU<2k> ! Z x BU *
*be the (2k - 1)-
connected cover, and let MU<2k> be the associated bordism theory. For an even p*
*eriodic ring spectrum E
and for k 3, the map
RingSpectra(MU<2k>; E) -!Algss0E(E0MU<2k>; ss0E)
is an isomorphism. In other words, the multiplicative maps MU<2k> ! E are in on*
*e-to-one correspondence
with ss0E-valued points of spec(ss0E ^ MU<2k>). If E is an elliptic spectrum, t*
*hen the Theorem of the Cube
endows this scheme with a canonical point. In order to connect the topology to *
*the algebraic geometry, we
shall express some facts about even periodic ring spectra in the language of al*
*gebraic geometry.
ELLIPTIC SPECTRA *
* 7
2.1.1. Formal schemes and formal groups. Following [DG70 ], we will think of an*
* affine scheme as a repre-
sentable covariant functor from rings to sets. The functor (co-)represented by *
*a ring A is denoted specA.
The ring (co-)representing a functor X will be denoted OX .
A formal scheme is a filtered colimit of affine schemes. For example, the fun*
*ctor bA1associating to a ring
R its set of nilpotent elements is the colimit of the schemes spec(Z[x]=xk) and*
* thus is a formal scheme.
The category of formal schemes has finite products: if X = colimXffand Y = co*
*limYfithen X x Y =
colimXffxYfi. The formal schemes in this paper will all be of the form bAnxZ = *
*bA1x: :x:bA1xZ for some
affine scheme Z. If X = colimffXffis a formal scheme, then we shall write OX fo*
*r limffOXff; in particular
we have ObA1= Z[[x]]. We write b for the completed tensor product, so that for *
*example
OXxY = OX bOY:
If X ! S is a morphism of schemes with a section j :S ! X, then bXwill denote*
* the completion of X
along the section. Explicitly, the section j defines an augmentation
*
OX -j!OS:
If J denotes the kernel of j*, then
bX= colimspec(OX =JN ):
N
For example, the zero element defines a section spec(Z) ! A1, and the completio*
*n of A1 along this section
is the formal scheme bA1.
A commutative one-dimensional formal group over S is a commutative group G in*
* the category of formal
schemes over S which, locally on S, is isomorphic to S x bA1as a pointed formal*
* scheme over S. We shall
often omit "commutative" and "one-dimensional", and simply refer to G as a form*
*al group.
We shall use the notation Ga for the additive group, and Gm for the multiplic*
*ative group. As functors we
have Ga(R) = R and Gm (R) = Rx. Thus bGais the additive formal group, and bGa(R*
*) is the additive group
of nilpotent elements of R.
If the group scheme Gm acts on a scheme X, we have a map ff:Gm x X ! X, corre*
*sponding to a map
ff*:OX -!OGmxX = OX [u; u-1]. We put (OX )n = {f | ff(f) = unf}. This makes OX *
*into a graded ring.
A graded ring R* is said to be of finite type over Z if each Rn is a finitely*
* generated abelian group.
2.1.2. Even ring spectra and schemes. If E is an even periodic ring spectrum, t*
*hen we write
SE def=spec(ss0E):
If X is a space, we write E0X and E0X for the unreduced E-(co)homology of X. *
*If A is a spectrum, we
write E0A and E0A for its spectrum (co)homology. These are related by the formu*
*la E0X = E01 X+.
Let X be a CW-complex. If {Xff} is the set of finite subcomplexes of X then w*
*e write XE for the formal
scheme colimffspec(E0Xff). This gives a covariant functor from spaces to formal*
* schemes over SE.
We say that X is even if H*X is a free abelian group, concentrated in even de*
*grees. If X is even and E
is an even periodic ring spectrum, then E0X is a free module over E0, and E0X i*
*s its dual. The restriction
to even spaces of the functor X 7! XE preserves finite products. For example th*
*e space P def=CP1 is even,
and PE is (non-canonically) isomorphic to the formal affine line. The multiplic*
*ation P x P ! P classifying
the tensor product of line bundles makes the scheme PE into a (one-dimensional *
*commutative) formal group
over SE.
The formal group PE is not quite the same as the one introduced by Quillen [Q*
*ui69]. The ring of functions
on Quillen's formal group is E*(P), while the ring of functions on PE is E0(P).*
* The homogeneous parts of
E*(P) can interpreted as sections of line bundles over PE. For example, let I b*
*e the ideal of functions on
PE which vanish along the identity section. The natural map
I=I2 ! "E0(S2) = ss2E (2*
*.1)
8 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
is an isomorphism. Now I=I2 is, by definition, the Zariski cotangent space to t*
*he group PE at the identity,
and defines a line bundle over specss0E. This line bundle is customarily denote*
*d !, and can be regarded as
the sheaf of invariant 1-forms on PE. In this way we will identify ss2E with in*
*variant 1-forms on PE. More
generally, ss2nE can be identified with the module of sections of !n (i.e., inv*
*ariant differentials of degree n
on PE).
Note that for any space X, the map
"E0(X) ss0Ess-2n(E) -!"E2n(X)
is an isomorphism, and so E2n(X) can be identified with the module of sections *
*of the pull-back of the line
bundle !-n to XE.
W
Let E be an even ring spectrum, which need not be periodic. Let EP = n2Z2n*
*E. There is an
evident way to make this into a commutative ring spectrum with the property tha*
*t ss*EP = E*[u; u-1] with
u 2 ss2EP. With thisLstructure, EP becomes an even periodic ring spectrum. Note*
* that when X is finite
we just have EP0X = nE2nX, so the ring EP0X has a natural even grading. If X *
*is an infinite, even
CW-complex then EP0X is the completed direct sum (with respect to the topology *
*defined by kernels of
restrictions to finite subcomplexes) of the groups E2nX and so again has a natu*
*ral even grading.
We write HP for the 2-periodic integer Eilenberg-MacLane spectrum HZP, and MP*
* for MUP = MU<0>.
The formal group of HP is the additive group bGa; and we may choose an additive*
* coordinate z on bGafor
which u = dz. By Quillen's theorem [Qui69], the formal group of MP is Lazard's *
*universal formal group
law.
If X is an even, homotopy commutative H-space, then XE is a (commutative but *
*in general not one-
dimensional) formal group. In that case E0X is a Hopf algebra over E0 and we wr*
*ite XE = spec(E0X) for
the corresponding group scheme. It is the Cartier dual of the formal group XE. *
*We recall (from [Dem72,
xII.4], for example; see also [Str99a, Section 6.4] for a treatment adapted to *
*the present situation) that the
Cartier dual of a formal group G is the functor from rings to groups
Hom_(G; Gm )(A) = {(u; f) | u: spec(A) -!S ; f 2 (Formal groups)(u*G;*
* u*Gm )}:
Let b 2 E0X bE0X be the adjoint of the identity map E0X ! E0X. Given a ring hom*
*omorphism g :E0X !
A we get a map u: spec(A) ! SE and an element g(b) 2 (Ab E0X)x = (Ab OXE)x, whi*
*ch corresponds to
a map of schemes
f :u*XE -!u*Gm :
One shows that it is a group homomorphism, and so gives a map of group schemes
XE -!Hom_(XE; Gm ); (2*
*.2)
which turns out to be an isomorphism.
2.2. Constructions of elliptic spectra. Recall that an elliptic spectrum is a t*
*riple (E; C; t) consisting of
an even, periodic, homotopy commutative ring spectrum E, a generalized elliptic*
* curve C over E0(pt), and
an isomorphism formal groups
t:PE -!Cb:
Here are some examples.
Example 2.3.As discussed in the introduction, if C is a lattice, then the quo*
*tient C= is an elliptic
curve C over C. The covering map C ! C= gives an isomorphism t :Cb ~=bGa. Let *
*E be the spectrum
representing the cohomology theory H*( - ; C[u ; u-1]). Define H to be the ell*
*iptic spectrum (E ; C ; t ).
Note that u can be taken to correspond to the invariant differential dz on C u*
*nder the isomorphism (2.1).
Given a non-zero complex number , consider the map
f :E -!E
u 7! u
ELLIPTIC SPECTRA *
* 9
(i.e. ss2f scales the invariant differential by ). The induced map of formal gr*
*oups is simply multiplication
by , and so extends to the isomorphism
C .-!C
of elliptic curves. Thus f defines a map of elliptic spectra
f :H -!H :
Example 2.4.Let CHP be the cuspidal cubic curve y2z = x3 over spec(Z). In xB.1*
*.4, we give an iso-
morphism s: (CHP )reg~=Ga and so ^s:bCHP~=bGa= PHP . Thus the triple (HP; CHP *
*; ^s) is an elliptic
spectrum.
Example 2.5.Let C = CK be the nodal cubic curve y2z + xyz = x3 over spec(Z). In*
* xB.1.4, we give an
isomorphism t:(CK )reg~=Gm so bCK~=bGm= PK . The triple (K; CK ; ^t) is an elli*
*ptic spectrum.
Example 2.6.Let C=S be an untwisted generalized elliptic curve (see Definition *
*B.2) with the property that
the formal group bCis Landweber exact (For example, this is automatic if OS is *
*a Q-algebra). Landweber's
exact functor theorem gives an even periodic cohomology theory E*( - ), togethe*
*r with an isomorphism of
formal groups t:PE -!Cb. This is the classical construction of elliptic cohomol*
*ogy; and gives rise to many
examples. In fact, the construction identifies a representing spectrum E up to *
*canonical isomorphism, since
Franke [Fra92] and Strickland [Str99a, Proposition 8.43] show that there are no*
* phantom maps between
Landweber exact elliptic spectra.
Example 2.7.In x2.6, we describe an elliptic spectrum based on the Tate ellipti*
*c curve, with underlying
spectrum K[[q]].
2.3. The complex-orientable homology of BU<2k> for k 3. Let E be an even perio*
*dic ring spectrum
with a coordinate x 2 eE0P, giving rise to a formal group law F over E0. Let ae*
*:P3 ! BU<6> be the map
(see (2.24)) such that the composition
P3 ae-!BU<6> ! BU
Q
classifies the virtual bundle i(1 - Li). Let f = f(x1; x2; x3) be the power se*
*ries which is the adjoint of E0ae
in the ring E0P3b E0BU<6> ~=E0BU<6>[[x1; x2; x3]]. It is easy to check that f s*
*atisfies the following three
conditions.
f(x1; x2;=0)1 (2.*
*8a)
f(x1; x2;ix3)s symmetric in the xi (2.*
*8b)
f(x1; x2; x3)f(x0; x1+F=x2;fx3)(x0+F x1; x2; x3)f(x0; x1; x3):(*
*2.8c)
We will eventually prove the following result.
Theorem 2.9.E0BU<6> is the universal example of an E0-algebra R equipped with a*
* formal power series
f 2 R[[x1; x2; x3]] satisfying the conditions (2.8).
In this section we will reformulate this statement (as the case k = 3 of Theo*
*rem 2.29) in a way which
avoids the choice of a coordinate.
2.3.1. The functor Ck.
Definition 2.10.If A and T are abelian groups, we let C0(A; T) be the group
C0(A; T) def=(Sets)(A; T);
and for k 1 we let Ck(A; T) be the subgroup of f 2 (Sets)(Ak; T) such that
f(a1; : :;:ak-1;=0)0; (2.1*
*1a)
f(a1; : :;:ak)is symmetric;in the ai (2.1*
*1b)
f(a1; a2; a3; : :;:ak) + f(a0; a1+ a2;=a3;f:(:;:ak)a0+ a1; a2; a3; : :;:ak)(+2*
*f(a0;.a1;1a3;1:c:;:ak):)
10 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
We refer to (2.11c)as the cocycle condition for f. It really only involves th*
*e first two arguments of f,
with the remaining arguments playing a dummy r^ole. Of course, because f is sym*
*metric, we have a similar
equation for any pair of arguments of f.
Remark 2.12. We leave it to the reader to verify that the condition (2.11a)can *
*be replaced with the weaker
condition
f(0; : :;:0) = 0 (2.1*
*1a')
Remark 2.13. Let Z[A] denote the group ring of A, and let I[A] be its augmentat*
*ion ideal. For k 0 let
Ck(A) def=SymkZ[A]I[A]
be the kthsymmetric tensor power of I[A], considered as a module over the group*
* ring. One has C0(A) = Z[A]
and C1(A) = I[A]. For k 1, the abelian group Ck(A) is the quotient of SymkZI[A*
*] by the relation
([c] - [c + a1]) ([0] - [a2]) : : :([0] - [ak]) = ([0] - [a1]) ([c] - [*
*c + a2]) : : :([0] - [ak])
for c 2 A. After some rearrangement and reindexing, this relation may be expres*
*sed in terms of generators
of the form def=([0] - [a1]) : : :([0] - [ak]) by the formula
- + - = 0:
It follows that the map of sets
Ak! Ck(A)
(a1; : :;:ak)7!
induces an isomorphism
(Abelian groups)(Ck(A); T) ~=Ck(A; T):
Remark 2.14. Definition 2.10 generalizes to give a subgroup Ck(A; B) of the gro*
*up of maps f :Ak -!B,
if A and B are abelian groups in any category with finite products.
Definition 2.15.If G and T are formal groups over a scheme S, and we wish to em*
*phasize the r^ole of S,
we will write CkS(G; T). For any ring R, we define
C_k(G; T)(R) = {(u; f) | u: spec(R) -!S ; f 2 Ckspec(R)(u*G; u*T*
*)}:
This gives a covariant functor from rings to groups. We shall abbreviate C_k(G;*
* Gm x S) to C_k(G; Gm ).
Remark 2.16. It is clear from the definition that, for all maps of schemes S0! *
*S, the natural map
C_k(G xS S0; Gm ) ! C_k(G; Gm ) xS S0
is an isomorphism.
Proposition 2.17.Let G be a formal group over a scheme S. For all k, the functo*
*r C_k(G; Gm ) is an affine
commutative group scheme.
Proof.We assume that k > 0, leaving the modifications for the case k = 0 to the*
* reader. It suffices to
work locally on S, and so we may choose a coordinate x on G. Let F be the resul*
*ting formal group law
of G. We let A be the set of multi-indices ff = (ff1; :P:;:ffk), where each ffi*
*is a nonnegative integer. We
define R = OS[bff| ff 2 A][b-10], and f(x1; : :;:xk) = ffbffxff2 R[[x1; : :;:*
*xk]]. Thus, f defines a map
spec(R) xS Gk -!Gm , and in fact spec(R) is easily seen to be the universal exa*
*mple of a scheme over S
equipped with such a map. We define power series g0; : :;:gk by
8
>*i < k f(x1; : :;:xi-1; xi+1; xi; : :;:xk)f(x1; : :;:xk)-1
:i = k f(x1; : :;:xk)f(x0+F x1; x2; : :):-1f(x0; x1+F x2; : :):f(x0;*
* x1; x3; : :):-1
We then let I be the ideal in R generated by all the coefficients of all the po*
*wer series gi- 1. It_is_not hard
to check that spec(R=I) has the universal property that defines C_k(G; Gm ). *
* |__|
ELLIPTIC SPECTRA *
* 11
Remark 2.18. A similar argument shows that C_k(G; T) is a group scheme when T i*
*s a formal group, or
when T is the additive group Ga.
Remark 2.19. If G is a formal group and k > 0 then the inclusion C_k(G; bGm) -!*
*C_k(G; Gm ) is an isomor-
phism, so we shall not distinguish between these two schemes. Indeed, we can lo*
*cally identify C_k(G; Gm )(R)
with a set of power series f as in the above proof. One of the conditions on f *
*is that f(0; : :;:0) = 1, so
when x1; : :;:xk are nilpotent we see that f(x1; : :;:xn) = 1 mod nilpotents, s*
*o f(x1; : :;:xn) 2 bGm Gm .
This does not work for k = 0, as then we have
C0(G; Gm ) = Map(G; Gm ) 6= Map(G; bGm) = C0(G; bGm):
2.3.2. The maps ffi :Ck(G; T) ! Ck+1(G; T). We now define maps of schemes that*
* will turn out to corre-
spond to the maps BU<2k + 2> -!BU<2k> of spaces.
Definition 2.20.If G and T are abelian groups, and if f :Gk ! T is a map of set*
*s, then let ffi(f):Gk+1! T
be the map given by the formula
ffi(f)(a0; : :;:ak) = f(a0; a2; : :;:ak) + f(a1; a2; : :;:ak) - f(a0*
*+(a1;2a2;.:2:;:ak):1)
It is clear that ffi generalizes to abelian groups in any category with produ*
*cts. We leave it to the reader
to verify the following.
Lemma 2.22. For k 1, the map ffi induces a homomorphism of groups
ffi :Ck(G; T) ! Ck+1(G; T):
Moreover, if G and T are formal groups over a scheme S, then ffi induces a homo*
*morphism of_group schemes
ffi :C_k(G; T) -!C_k+1(G; T). *
* |__|
Remark 2.23. When A and T are discrete abelian groups, the group H2(A; T) def=c*
*ok(ffi :C1(A; T) -!
C2(A; T)) classifies central extensions of A by T. The next map ffi :C2(A; T) *
*-! C3(A; T)) can also be
interpreted in terms of biextensions [Mum65 , Gro72, Bre83].
2.3.3. Relation to BU<2k>. For any space X, we write K*(X) for the periodic com*
*plex K-theory groups of
X; in the case of a point we have K* = Z[v; v-1] with v 2 K-2. We have K2t(X) =*
* [X; Z x BU] for all
t. We also consider the connective K-theory groups bu*(X), so bu* = Z[v] and bu*
*2t(X) = [X; BU<2t>]. To
make this true when t = 0, we adopt the convention that BU<0> = Z x BU. Multipl*
*ication by vt:2tbu !
bu gives an identification of the 0-space of 2tbu with BU<2t>. Under this iden*
*tification, the projection
BU<2t + 2> -!BU<2t> is derived from multiplication by v mapping 2t+2bu ! 2tbu.
For t 0 we define a map
aet:Pt = (CP1 )t! BU<2t> (2.*
*24)
as follows. The map ae0:P ! 1 x BU BU<0> is just the map classifying the tauto*
*logical line bundle L.
For t > 0, let L1; : :;:L2 be the obvious line bundles over Pt. Let xi2 bu2(Pt)*
* be the bu-theory Euler class,
given by the formula
vxi= 1 - Li:
Then we have the isomorphisms
bu*(Pt)~=Z[v][[x1; : :;:xt]]
K*(Pt)~=Z[v; v-1][[x1; : :;:xt]]:
Q
The class ixi2 bu2t(Pt) gives the map aet. Note that the composition
Pt aet-!BU<2t> ! BU
Q
classifies the bundle i(1 - Li).
Since P and BU<2t> are abelian group objects in the homotopy category of topo*
*logical spaces, we can
define
Ct(P; BU<2t>) [Pt; BU<2t>] = bu2t(Pt):
12 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
Then we have the following.
Proposition 2.25.The map aetis contained in the subgroup Ct(P; BU<2t>) of bu2t(*
*Pt) and satisfies
v*aet+1= ffi(aet) 2 Ct+1(P; BU<2t>):
Proof.It suffices to check that ftgives an element of Ct(P; BU<0>). As the grou*
*p structure of P corresponds
to the tensor product of line bundles, while the group structure of BU<0> corre*
*sponds to the Whitney sum
of vector bundles, the cocycle condition (2.11c)amounts to the equation
(1 - L2)(1 - L3) + (1 - L1)(1 - L2L3) = (1 - L1L2)(1 - L3) + (1 - L1)*
*(1 - L2)
in K0(P3). The other conditions for membership in Ct are easily verified. Simil*
*arly, the equation v*aet+1=
ffi(aet) follows from the equation
(1 - L1) + (1 - L2) - (1 - L1L2) = (1 - L1)(1 - L2):
*
*|___|
Now let E be an even periodic ring spectrum. Applying E-homology to the map a*
*ek gives a homomorphism
E0aek:E0Pk ! E0BU<2k>:
For k 3, BU<2k> is even ([Sin68] or see x4), and of course the same is true of*
* P, and so we may consider
the adjoint ^aekof E0aek in E0BU<2k>b E0Pk. Proposition 2.25 then implies the f*
*ollowing.
Corollary 2.26.The element ^aek2 E0BU<2k>b E0Pk is an element of C_k(PE; Gm )(E*
*0BU<2k>). |___|
Definition 2.27.For k 3, let fk:BU<2k>E ! C_k(PE; Gm ) be the map classifying *
*the cocycle ^aek.
Corollary 2.28.The map fk is a map of group schemes. For k 2, the diagram
E
BU<2k + 2>E_v___//_BU<2k>E
fk+1|| |fk|
fflffl| fflffl|
C_k+1(PE; Gmf)fi//_C_k(PE; Gm )
commutes.
Proof.The commutativity of the diagram follows easily from the Proposition. To *
*see that fk is a map of
group schemes, note that the group structure on BU<2k>E is induced by the diago*
*nal map :BU<2k> -!
BU<2k> x BU<2k>. The commutative diagram
Pk ----! Pk x Pk
? ?
aek?y ?yaekxaek
BU<2k> ----! BU<2k> x BU<2k>
shows that
BU<2k>E x BU<2k>E -!BU<2k>E
pulls the function ^aekback to the multiplication of ^aek1 and 1^aekas elements*
* of the ring E0(BU<2k>2)b E0Pk
of functions on PkEx (BU<2k>E x BU<2k>E). The result follows, since the group s*
*tructure of C_k(PE;_Gm )
is induced by the multiplication of functions in OPkE. *
* |__|
Our main calculation, and the promised coordinate-free version of Theorem 2.9*
*, is the following.
Theorem 2.29.For k 3, the map of group schemes
BU<2k>E fk-!C_k(PE; Gm )
is an isomorphism.
ELLIPTIC SPECTRA *
* 13
This is proved in x4. The cases k 1 are essentially well-known calculations.*
* For k = 2 and k = 3
we can reduce to the case E = MP, using Quillen's theorem that ss0MP carries th*
*e universal example of
a formal group law. Using connectivity arguments and the Atiyah-Hirzebruch spec*
*tral sequence, we can
reduce to the case E = HP. After these reductions, we need to compare H*BU<2k> *
*with OC_k(bGa;Gm). We
analyze H*(BU<2k>; Q) and H*(BU<2k>; Fp) using the Serre spectral sequence, and*
* we analyze OC_k(bGa;Gm)
by direct calculation, one prime at a time. For the case k = 3 we also give a m*
*odel for the scheme associated
to the polynomial subalgebra of H*(K(Z; 3); Fp), and by fitting everything toge*
*ther we show that the map
BU<2k>E -!C_k(PE; Gm ) is an isomorphism.
Remark 2.30. As BU<2k>E = Hom_(BU<2k>E; Gm ) = C_k(PE; Gm ), it is natural to h*
*ope that one could
i.define a formal group scheme Ck(PE) which could be interpreted as the k'th *
*symmetric tensor power of
the augmentation ideal in the group ring of the formal group PE;
ii.show that C_k(PE; Gm ) = Hom_(Ck(PE); Gm ); and
iii.prove that BU<2k>E = Ck(PE).
This would have advantages over the above theorem, because the construction X 7*
*! XE is functorial for
all spaces and maps, whereas the construction X 7! XE is only functorial for co*
*mmutative H-spaces and
H-maps. It is in fact possible to carry out this program, at least for k 3. *
*It relies on the apparatus
developed in [Str99a], and the full strength of the present paper is required e*
*ven to prove that C3(G) (as
defined by a suitable universal property) exists. Details will appear elsewhere.
2.4. The complex-orientable homology of MU<2k> for k 3. We now turn our attent*
*ion to the
Thom spectra MU<2k>. We first note that when k 3, the map BU<2k> ! BU<0> = Z x*
* BU is a map
of commutative, even H-spaces. The Thom isomorphism theorem as formulated by [M*
*R81 ] implies that
E0MU<2k> is an E0BU<2k>-comodule algebra; and a choice of orientation MU<0> ! E*
* gives an isomorphism
E0MU<2k> ~=E0BU<2k>
of comodule algebras. In geometric language, this means that the scheme MU<2k>E*
* is a principal homoge-
neous space or "torsor" for the group scheme BU<2k>E.
In this section, we work through the Thom isomorphism to describe the object *
*which corresponds to
MU<2k>E under the isomorphism BU<2k>E ~=C_k(PE; Gm ) of Theorem 2.29. Whereas t*
*he schemes BU<2k>E
are related to functions on the formal group PE of E, the schemes MU<2k>E are r*
*elated to the sections of
the ideal sheaf I(0) on PE. In x2.4.4, we describe the analogue C_k(G; I(0)) fo*
*r the line bundle I(0) of the
functor C_k(G; Gm ). In x2.4.5, we give the map
gk:MU<2k>E ! C_k(PE; I(0))
which is our description of MU<2k>E.
2.4.1. Torsors. We begin with a brief review of torsors in general and the Thom*
* isomorphism in particular.
Definition 2.31.Let S be a scheme and G a group scheme over S. A (right) G-tors*
*or over S is an S-scheme
X with a right action
X x G -!X
of the group G, with the property that there exists a faithfully flat S-scheme *
*T and an isomorphism
G x T -!X x T
of T-schemes, compatible with the action of G x T. (All the products here are t*
*o be interpreted as fiber
products over S.) Any such isomorphism is a trivialization of X over T. A map o*
*f G-torsors is just an
equivariant map of schemes. Note that a map of torsors is automatically an isom*
*orphism.
When G = spec(H) is affine over S = spec(A), a G-torsor works out to consists*
* of an affine S-scheme
T = spec(M) and a right coaction
*
M -! M A H
14 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
with the property that over some faithfully flat A-algebra B there is an isomor*
*phism
H A B -!M A B
of rings which is a map of right H A B-comodules.
For example, consider the relative diagonal
MU<2k> -!MU<2k> ^ BU<2k>+:
If E is an even periodic ring spectrum and k 3, then by the K"unneth and unive*
*rsal coefficient theorems,
the map induces an action
MU<2k>E x BU<2k>E -!MU<2k>E:
of the group scheme BU<2k>E on MU<2k>E. The scheme MU<2k>E is in fact a torsor *
*for BU<2k>E. Indeed, a
complex orientation MU<0> -!E restricts to an orientation :MU<2k> -!E which ind*
*uces an isomorphism
E0MU<2k> -!E0MU<2k> ^ BU<2k>+ ^BU<2k>+--------!E0BU<2k>+ (2.*
*32)
of E0BU<2k>-comodule algebras.
2.4.2. The line bundle I(0). Another source of torsors is line bundles. If L is*
* a line bundle (invertible sheaf
of OX -modules) over X, let x(L) be the functor of rings
x(L)(R) = {(u; s) | u: spec(R) -!X ; s a trivializationuof*L}:
Then x(L) is a Gm -torsor over X, and x is an equivalence between the category *
*of line bundles (and
isomorphisms) and the category of Gm torsors. We will often not distinguish in *
*notation between L and the
associated Gm -torsor x(L).
Let G be a formal group over a scheme S. The ideal sheaf I(0) associated to t*
*he zero section S G
defines a line bundle over G. Indeed, the set of global sections of I(0) is the*
* set of functions f 2 OG such
that f|S = 0. Locally on S, a choice of coordinate x gives an isomorphism OG = *
*OS[[x]], and the module of
sections is the ideal (x), which is free of rank 1.
If C is a generalized elliptic curve over, then we again let I(0) denote the *
*ideal sheaf of S C. Its
restriction to the formal completion bCis the same as the line bundle over bCco*
*nstructed above.
2.4.3. The Thom sheaf. Suppose that X is a finite complex and V is a complex ve*
*ctor bundle over X.
We write XV for its Thom spectrum, with bottom cell in degree equal to the real*
* rank of V . This is the
suspension spectrum of the usual Thom space. Now let E be an even periodic ring*
* spectrum. The E0X-
module E0XV is the sheaf of sections of a line bundle over XE. We shall write L*
*(V ) for this line bundle,
and L defines a functor from vector bundles over X to line bundles over XE. If *
*V and W are two complex
vector bundles over X then there is a natural isomorphism
L(V W) ~=L(V ) L(W); (2.*
*33)
and so L extends to the category of virtual complex vector bundles by the formu*
*la L(V - W) = L(V )
L(W)-1. Moreover, if f :Y ! X is a map of spaces, then there is a natural isomo*
*rphism (specE0f)*L(V ) ~=
L(f*V ) of line bundles over YE. This construction extends naturally to infinit*
*e complexes by taking suitable
(co)limits.
Example 2.34.For example, if L is the tautological line bundle over P = CP1 th*
*en the zero section
P -!PL induces an isomorphism eE0PL ~=eE0P = ker(E0P -!E0), and thus gives an i*
*somorphism
L(L) ~=I(0) (2.*
*35)
of line bundles over PE.
ELLIPTIC SPECTRA *
* 15
2.4.4. The functors k (after Breen [Bre83]). We recall that the category of lin*
*e bundles or Gm -torsors is
a strict Picard category, or in other words a symmetric monoidal category in wh*
*ich every object L has an
inverse L-1, and the twist map of L L is the identity. This means that the pro*
*cedures we use below to
define line bundles give results that are well-defined up to coherent canonical*
* isomorphism.
Suppose that G is a formal group over a scheme S and L is a line bundle over *
*G.
Definition 2.36.A rigid line bundle over G is a line bundle L equipped with a s*
*pecified trivialization of L|S
at the identity S ! G. A rigid section of such a line bundle is a section s whi*
*ch extends the specified section
at the identity. A rigid isomorphism between two rigid line bundles is an isomo*
*rphism which preserves the
specified trivializations.
Definition 2.37.SupposePthat k 1. Given a subset I {1; : :;:k}, we define oe*
*I:GkS-! G by
oeI(a1; : :;:ak) = i2Iai, and we write LI = oe*IL, which is a line bundle ove*
*r GkS. We also define the
line bundle k(L) over GkSby the formula
O |I|
k(L) def= (LI)(-1) : (2.*
*38)
I{1;:::;k}
Finally, we define 0(L) = L.
For example we have
0(L)a = La
1(L)a = L0_L
a
2(L)a;b= L0_La+b_L
a Lb
3(L)a;b;c= L0_La+b_La+c_Lb+c__L:
a Lb Lc La+b+c
We observe three facts about these bundles.
i.k(L) has a natural rigid structure for k > 0.
ii.For each permutation oe 2 k, there is a canonical isomorphism
oe: ss*oek(L) ~=k(L);
where ssoe:GkS-!GkSpermutes the factors. Moreover, these isomorphisms compo*
*se in the obvious way.
iii.There is a canonical identification (of rigid line bundles over Gk+1S)
k(L)a1;a2;::: k(L)-1a0+a1;a2;::: k(L)a0;a1+a2;::: k(L)-1a0;a1;:::*
*~=1:(2.39)
Definition 2.40.A k-structure on a line bundle L over a group G is a trivializa*
*tion s of the line bundle
k(L) such that
i.for k > 0, s is a rigid section;
ii.s is symmetric in the sense that for each oe 2 k, we have oess*oes = s;
iii.the section s(a1; a2; : :): s(a0+ a1; a2; : :):-1 s(a0; a1+ a2; : :): s(a0*
*; a1; : :):-1corresponds to 1
under the isomorphism (2.39).
A 3-structure is known as a cubical structure [Bre83]. We write Ck(G; L) for t*
*he set of k-structures
on L over G. Note that C0(G; L) is just the set of trivializations of L, and C*
*1(G; L) is the set of rigid
trivializations of 1(L). We also define a functor from rings to sets by
C_k(G; L)(R) = {(u; f) | u: spec(R) -!S ; f 2 Ckspec(R)(u*G; u*L*
*)}:
Remark 2.41. Note that for the trivial line bundle OG, the set Ck(G; OG) reduce*
*s to that of the group
Ck(G; Gm ) of cocycles introduced in x2.3.1.
16 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
Remark 2.42. There are some differences between our functors k and Breen's func*
*tors and [Bre83].
Let L0= 1(L)-1 be the line bundle La=L0: Then there are natural isomorphisms
(L0) ~=2(L)
(L0) ~=3(L)-1:
Breen also uses the notation 1(M) for (L0) [Bre83, Equation 2.8.1]. As the triv*
*ializations of L biject with
those of L-1 in an obvious way, our definition of cubical structures is equival*
*ent to Breen's.
Proposition 2.43.If G is a formal group over S, and L is a trivializable line b*
*undle over G, then the
functor C_k(G; L) is a scheme, whose formation commutes with change of base. Mo*
*reover, C_k(G; L) is a
trivializable torsor for C_k(G; Gm ).
Proof.There is an evident action of C_k(G; Gm ) on C_k(G; L), and a trivializat*
*ion of L clearly gives an
equivariant isomorphism of C_k(G; L) with C_k(G; OG) = C_k(G; Gm ). Given this,*
* the Proposition follows
from the corresponding statements for C_k(G; Gm ), which were proved in Proposi*
*tion 2.17. |___|
The following lemmas can easily be checked from Definitions 2.37 and 2.40.
Lemma 2.44. If L is a line bundle over a formal group G, then there is a canoni*
*cal isomorphism
k(L)a0;a2;::: k(L)a1;a2;::: k(L)-1a0+a1;a2;:::~=k+1(L)a0;:::;ak: *
* |___|
Lemma 2.45. There is a natural map ffi :C_k(G; L) -!C_k+1(G; L), given by
ffi(s)(a0; : :;:ak) = s(a0; a2; : :):s(a1; a2; : :):s(a0+ a1; a*
*2; : :):-1;
where the right hand side is regarded as a section of k+1(L) by the isomorphism*
* of the previous lemma. |___|
2.4.5. Relation to MU<2k>. For 1 i k, let Libe the line bundle over the i fac*
*tor of Pk. Recall from
(2.24)that the map aek:Pk ! BU<2k> pulls the tautological virtual bundle over B*
*U<2k> back to the bundle
O
V = (1 - Li):
i
Passing to Thom spectra gives a map
(Pk)V ! MU<2k>
which determines an element sk of E0MU<2k>b E0((Pk)V).
We recall from (2.35)that there is an isomorphism of line bundles L(L) ~=I(0)*
* over PE, where I(0) is
the ideal sheaf of the zero section; and that the functor L (from virtual vecto*
*r bundles to line bundles over
XE) sends direct sums to tensor products. Together these observations give an i*
*somorphism
L(V ) ~=k(I(0)) (2.*
*46)
of line bundles over PkE. With this identification, sk is a section of the pul*
*l-back of k(I(0)) along the
projection MU<2k>E -!SE.
Lemma 2.47. The section sk is a k-structure.
Proof.This is analogous to Corollary 2.26. *
* |___|
Let
MU<2k>E gk-!C_k(PE; I(0))
be the map classifying the k-structure sk. We note that the isomorphism BU<2k>E*
* ~=C_k(PE; Gm ) gives
C_k(PE; I(0)) the structure of a torsor for the group scheme BU<2k>E.
Theorem 2.48.For k 3, the map gk is a map of torsors for the group BU<2k>E (an*
*d so an isomorphism).
Moreover, the map MU<2k + 2> -!MU<2k> induces the map ffi :C_k(PE; I(0)) -!C_k+*
*1(PE; I(0)).
ELLIPTIC SPECTRA *
* 17
Proof.Let us write for the action
C_k(PE; I(0)) x C_k(PE; Gm ) -!C_k(PE; I(0)):
If funivis the universal element of C_k(PE; Gm ) and sunivis the universal elem*
*ent of C_k(PE; I(0)), then
is characterized by the equation
*suniv= funivsuniv; (2.*
*49)
as elements of C_k(PE; I(0))(OC_k(PE;I(0))xC_k(PE;Gm)).
Now consider the commutative diagram
(Pk)V_______//_(Pk)V ^ (Pk)+
| |
| |
fflffl| fflffl|
MU<2k> ____//_MU<2k> ^ BU<2k>+:
Applying E-homology and then taking the adjoint in E0(BU<2k>+ ^ MU<2k>)b E0(Pk)*
*V gives a section
of k(I(0)) over BU<2k>E x MU<2k>E. The counterclockwise composition identifies*
* this section as the
pull-back of the section sk under the action
E
MU<2k>E x BU<2k>E --! MU<2k>E
as in x2.4.1. Via the isomorphism BU<2k>E ~=C_k(PE; Gm ) of Theorem 2.29, the c*
*lockwise composition is
funivsk. From the description of (2.49)it follows that gk is a map of torsors,*
* as required.
Another diagram chase shows that the map MU<2k + 2> -! MU<2k> is compatible w*
*ith the map
ffi :C_k(GE; I(0)) -!C_k+1(GE; I(0)). *
* |___|
Corollary 2.50.For 0 k 3, maps of ring spectra MU<2k> -!E are in bijective co*
*rrespondence with
k-structures on I(0) over GE.
Proof.Since E*MU<2k> is torsion free and concentrated in even degrees, one has
[MU<2k>; E] = E0MU<2k> = Homss0E(E0MU<2k>; ss0E):
One checks that maps of ring spectra correspond to ring homomorphisms, so
RingSpectra(MU<2k>; E) = Algss0E(E0MU<2k>; ss0E):
This is just the set of global sections of MU<2k>E over SE, which is the set of*
* k-structures_on I(0) over
GE by the theorem. *
* |__|
Example 2.51.Maps of ring spectra MP = MU<0> -! E are in bijective corresponden*
*ce with global
trivializations of the sheaf I(0) ~=L(L), that is, with generators x of the aug*
*mentation ideal E0P ! E0(pt).
Example 2.52.Maps of ring spectra MU = MU<2> ! E are in bijective correspondenc*
*e with rigid sections
of ! I(0)-1, or equivalently with rigid sections of !-1I(0). The isomorphism (2*
*.46)identifies sections of
!-1 I(0) with elements of E0(PL-1), and the rigid sections are those which rest*
*rict to the identity under
the inclusion
S0 ! PL-1
of the bottom cell. It is equivalent to give a class x 2 "E2(P) whose restricti*
*on to "E2(S2) is the suspension
of 1 2 "E0S0; this is the description of maps MU ! E in [Ada74].
2.5. The oe-orientation of an elliptic spectrum.
18 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
2.5.1. Elliptic spectra and the Theorem of the Cube. Let C be a generalized ell*
*iptic curve over an affine
scheme S. To begin, note that the smooth locus Cregis a group scheme over S, so*
* we can define 3(I(0))
over Creg. We define a cubical structure on C to be a cubical structure on I(0)*
*|Creg; and we write C_3(C; I(0))
for C_3(Creg; I(0)).
Theorem 2.53.For any (nonsingular) elliptic curve C over a normal scheme S, the*
*re is a unique cubical
structure s(C=S) 2 C_3(C; I(0)). It has the following properties:
i.If C0=S0is obtained from C=S by base change along f :S0-! S, then
s(C0=S0) = f*s(C=S)
ii.If t:C0-! C is an isomorphism over S, then
s(C0=S) = (t3)*s(C=S):
Proof.The first claim follows from [Gro72, Expose VIII, Cor. 7.5] (see also [Br*
*e83, Proposition 2.4]);_the
argument was sketched in the introduction. The other claims are immediate by un*
*iqueness. |__|
We would like to extend this to the case where S need not be normal and C is *
*allowed to have singularities.
In this generality there may be many cubical structures (for example when C is *
*a cuspidal cubic over spec(Z),
with Creg= Ga) but nonetheless there will be a canonical choice of one. To prov*
*e this, we will exhibit a
formula which gives the unique cubical structure on the universal elliptic curv*
*e over Z[a1; a2; a3; a4; a6][-1]
and give a density argument to show that this formula works in general.
Definition 2.54.Let C = C(a1; a2; a3; a4; a6) be a Weierstrass curve (see Appen*
*dix B for definitions and
conventions). A typical point of (Creg)3Swill be written as (c0; c1; c2). We *
*define s(a_) by the following
expression:
fifi fi-1
fix0 y0 z0fifififix0z0fifififix1z1fifififix2z2fifi
s(a_)(c0; c1; c2) = fifix1y1z1fifififi fifififififififififi(z0z1z2)-1d*
*(x=y)0:
fix2 y2 z2fi x1 z1 x2 z2 x0 z0
(Compare [Bre83, Equation 3.13.4], bearing in mind the isomorphism x 7! ["(x) :*
* "0(x) : 1] from C= to E;
Breen cites [FS80, Jac] as sources.)
Proposition 2.55.s(a_) is a meromorphic section of the line bundle p*!C over (C*
*reg)3S(where p:C3S-!S
is the projection). It defines a rigid trivialization of
(p*!C) I-D1+D2-D3= 3(I(0))
(in the notation of xB.4.2).
The proof is given in xB.4 of the appendix.
Corollary 2.56.There is a unique way to assign to a generalized elliptic curve *
*C over a scheme S a cubical
structure s(C=S) 2 C_3(C; I(0)), such that the following conditions are satisfi*
*ed.
i.If C0=S0is obtained from C=S by base change along f :S0-! S, then
s(C0=S0) = f*s(C=S)
ii.If t:C0-! C is an isomorphism over S, then
s(C0=S) = (t3)*s(C=S):
Proof.Over the locus WCell WC where is invertible, there is only one rigid tri*
*vialization of 3(I(0)),
and it is a cubical structure (by Theorem 2.53). Thus s(a_) satisfies the equat*
*ions for a cubical structure
when restricted to the dense subscheme C3regxWC WCell C3reg, so it must satisfy*
* them globally. Similarly,
the uniqueness clause in the theorem implies that s(a_)|WCellis invariant under*
* the action of the group WR,
and thus s(a_) itself is invariant.
Now suppose we have a generalized elliptic curve C over a general base S. At *
*least locally, we can choose
a Weierstrass parameterization of C and then use the formula s(a_) to get a cub*
*ical structure. Any other
ELLIPTIC SPECTRA *
* 19
Weierstrass parameterization is related to the first one by the action of WR, s*
*o it gives the same cubical
structure by the previous paragraph. We can thus patch together our local cubic*
*al structures to get_a global
one. The stated properties follow easily from the construction. *
* |__|
Theorem 2.57.For any elliptic spectrum E = (E; C; t) there is a canonical map o*
*f ring spectra
oeE :MU<6> -!E:
This map is natural in the sense that if f :E -!E0= (E0; C0; t0) is a map of el*
*liptic spectra, then the diagram
MU<6>G
oeExxxx GGoeE0GG
xx GGG
--xxx ##G
E _______f______//_E0
commutes (up to homotopy).
Proof.This is now very easy. Let s(C=SE) be the cubical structure constructed i*
*n Corollary 2.56, and let
s(Cb=SE) be the restriction of s(C=S) to bCE. The orientation is the map oeE : *
*MU<6> -!E corresponding
to t*s(Cb=S) via Corollary 2.50. The functoriality follows from the functoriali*
*ty of s in the corollary. |___|
2.6. The Tate curve. In this section we describe the Tate curve CTate, and give*
* an explicit formula for the
cubical structure s(CbTate). For further information about the Tate curve, the *
*reader may wish to consult for
example [Sil94, Chapter V] or [Kat73].
By way of motivation, let's work over the complex numbers. Elliptic curves ov*
*er C can be written in the
form
Cx=(u ~ qu)
for some q with 0 < |q| < 1. This is the Tate parameterization, and as is custo*
*mary, we will work with all q
at once by considering the family of elliptic curves
C0an=D0= D0x Cx=(q; u) ~ (q; qu);
parameterized by the punctured open unit disk
D0= {q 2 C | 0 < |q| < 1}:
In this presentation, meromorphic functions on C0anare naturally identified wit*
*h meromorphic functions
f(q; u) on D0x Cx satisfying the functional equation
f(q; qu) = f(q; u): (2.*
*58)
Sections of line bundles on C0anadmit a similar description, but with (2.58)mod*
*ified according to the descent
datum of the line bundle.
Let I(0) be the ideal sheaf of the origin on C0an. The pullback of I(0) to D0*
*x Cx is the line bundle whose
holomorphic sections are functions vanishing at the points (q; qn), with n 2 Z.*
* One such function is
"(q; u) = (1 - u) Y (1 - qnu)(1 - qnu-1);
n>0
which has simple zeroes at the powers of q, and so gives a trivialization of th*
*e pullback of I(0) to Cx.
The function "(q; u) does not descend to a trivialization of I(0) on C0an, but *
*instead satisfies the functional
equation
"(q; qu) = -u-1"(q; u): (2.*
*59)
However, as one can easily check,
ffi3"(q; u)
does descend to a rigid trivialization of 3(I(0)), and hence gives the unique c*
*ubical structure.
20 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
The curve C0anhas the following presentation as a Weierstrass curve. Set
X
oek(n)= dk
d|n
X
ffk= oek(n)qn
n>0
a4= -5ff3
a6= -(5ff3+ 7ff5)=12
(The coefficients of a6 are in fact integers). Consider the Weierstrass cubic
y2+ xy = x3+ a4x + a6 (2.*
*60)
over D0.
Proposition 2.61.The formulae
X X
x = __u___(1+- u)2qn d(ud- 2 + u-d)
n>0 d|n
2 X X d
y = __u___(1+- u)3qn _ ((d - 1)ud+ 2 - (d + 1)u-d):
n>0 d|n2
give an analytic isomorphism between the projective plane curve defined by (2.6*
*0)and C0an.
Proof.See for example [Sil94, Chapter V x1]. *
* |___|
Equation (2.60)makes sense for q = 0 and defines a family Canof generalized e*
*lliptic curves over the open
unit disk
D = {q 2 C | |q| < 1}:
The fiber of Canover q = 0 is the twisted cubic curve
y2 = x3:
The invariant differential of Canis given by
__dx_ = du_:
2y + x u
By continuity and Corollary 2.56, the expression ffi3"(q; u) determines the cub*
*ical structure on Can.
Let A Z[[q]] be the subring consisting of power series which converge absolu*
*tely on the open unit disk
{q 2 C | |q|:< 1}
The series a4and a6are in fact elements of A, and so (2.60)defines a generalize*
*d elliptic curve C over specA.
The curve Canis obtained by change of base from A to the ring of holomorphic fu*
*nctions on D. The Tate
curve CTateis the generalized elliptic curve over
DTate= specZ[[q]]
obtained by change of base along the inclusion A Z[[q]]. Since the map from th*
*e meromorphic sections of
3(I(0)) on C3 to meromorphic sections on C3anis a monomorphism, one can interpr*
*et the expression
s(C3an) = ffi3"(q; u)
as a formula for the cubical structure on the sheaf I(0) over C, and thus by ba*
*se change, for CTate.
Now the map
D0x Cx = D0x Gm ! C0an
is a local analytic isomorphism, and restricts to an isomorphism of formal grou*
*ps
D0x bGm! bC0an:
ELLIPTIC SPECTRA *
* 21
This, in turn, extends to an analytic isomorphism
D x bGm! bCan: (2.*
*62)
Although "(q; u) does not descend to a meromorphic function on Can, it does ext*
*end to a function on the
formal completion bCan. In fact it can be taken to be a coordinate on bCan. We *
*have therefore shown
Proposition 2.63.The pullback of the canonical cube structure s(Can) to bC3an, *
*is given by
s(Cban) = ffi3"(q; u);
where "(q; u) is interpreted as a coordinate on bCanvia (2.62). *
* |___|
We now have three natural coordinates on bC0an:
t = x=y; "(q; u); and 1 - u:
Of these, only the function t gives an algebraic coordinate on C0an(and in fact*
* on Can). Let's write each of
the above as formal power series in t:
"(q; u) = "(t) = t + O(t2)
1 - u = 1 - u(t) = t + O(t2):
By definition, the coefficients of the powers of t in the series "(t) and u(t) *
*are holomorphic functions on
the punctured disc D0. It is also easy to check that they in fact extend to hol*
*omorphic functions on D (set
q = 0) and have integer coefficients (work over the completion of Z[u1 ][[q]] a*
*t (1 - u)). Thus "(t) and u(t)
actually lie in A[[t]], and in this way can be interpreted as functions on the *
*formal completion of bCof C (and
hence, after change of base, on the completion bCTateof CTate). The function 1 *
*- u(t) gives an isomorphism
sTatedef=1 - u(t):Cb! bGm (2.*
*64)
Moreover, the restriction of the cubical structure s(C) to bC3is given by
s(Cb) = ffi3"(t);
since the map from the ring of formal functions on bCto the ring of formal func*
*tions on bCanis a monomor-
phism. Thus we have proved
Proposition 2.65.The canonical cubical structure s(Cb=A) 2 C_3(Cb; I(0)) is giv*
*en by the formula
s(Cb=A) = ffi3"(t);
where t = x=y, and "(t) is the series defined above. *
* |___|
2.7. The elliptic spectrum KTateand its oe-orientation. The multiplicative coho*
*mology theory under-
lying KTateis simply K[[q]]; so ss0KTate= Z[[q]]. The formal group comes from t*
*hat of K-theory via the
inclusion
K ,! K[[q]];
and is just the multiplicative formal group. The elliptic curve is the Tate ell*
*iptic curve CTate. The triple
(K[[q]]; CTate; sTate) is the Tate elliptic spectrum, which we shall denote sim*
*ply KTate.
By Proposition 2.65 and Theorem 2.48, the oe-orientation is the composite
"
MU<6> ! MP -!K[[q]];
with the map labeled "corresponding to the coordinate "(t) on bCTatein the isom*
*orphism of Theorem 2.48.
In this section, we express the map
"
ss*MU ! ss*MP -ss*-!ss*K[[q]]
in terms of characteristic classes, and identify the corresponding bordism inva*
*riant with the Witten genus.
22 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
According to Theorem 2.48, maps
MP ! E
are in one-to-one correspondence with coordinates f on the formal group. The re*
*striction
MU ! MP ! E
sends the coordinate f to the rigid section ffif of 1(I(0)) = I(0)0 I(0)-1: Th*
*e most straightforward
formula for ffif is
ffif = f(0)_f
which can be misleading, because it is tempting to write f(0) = 0. (The point i*
*s that it is not so when
regarded as a section of I(0)0.) It seems clearer to express ffif in terms of t*
*he isomorphism
I(0)0 I(0)-1 ~=! I(0)-1
as in x2.1.2. Sections of ! can be identified with invariant one-forms on PE. I*
*f x is a coordinate on PE, and
f(x) is a trivialization of I(0), then
0(0)Dx
ffif = f______f(x)
where Dx is the invariant differential with value dx at 0.
The K-theory orientation of complex vector bundles
MP ! K (2.*
*66)
constructed by Atiyah-Bott-Shapiro [ABS64] corresponds to the coordinate 1 - u *
*on the formal completion
of Gm = specZ[u; u-1]. The invariant differential is
D(1 - u) = -du_u;
and the restriction of (2.66)to MU ! K is classified by the 1-structure
ffi(1 - u) = _1__1 --udu_u:
The map
"
MU ! MP -!KTate
factors as
0)
MU ! MU ^ BU+ ffi(1-u)^(--------!KTate;
where 0is the element of BUKTate~=C1(CbTate; Gm ) given by the formula
Y (1 - qn)2
0= _________________nn-1:
n1 (1 - q u)(1 - q u )
In geometric terms, the homotopy groups
ss*MU ^ BU+
are the bordism groups of pairs (M; V ) consisting of a stably almost complex m*
*anifold M, and a virtual
complex vector bundle V over M of virtual dimension 0. The map
ss*MU ! ss*MU ^ BU+
sends a manifold M to the pair (M; ) consisting of M and its reduced stable nor*
*mal bundle.
The map ss*ffi(1 - u) sends a manifold M of dimension 2n to
f!1 2 K-2n(pt) "K0(S2n);
ELLIPTIC SPECTRA *
* 23
where
f : M ! pt
is the unique map. One has
n
f!1 = Td(M) -du_u ;
where Td(M) is the Todd genus of M, and it is customary to suppress the grading*
* and write simply
f!1 = Td(M):
The map 0is the stable exponential characteristic class taking the value
Y (1 - qn)2
_________________nn-1
n1 (1 - q L)(1 - q L )
on the reduced class of a line bundle (1 - L). This stable exponential charact*
*eristic class can easily be
identified with
O
V 7! Symqn(-VC);
n1
where VC = V R C, VC= VC - CdimV, and Symt(W) is defined for (complex) vector b*
*undles W by
M
Symt(W) = Symn(V ) tn 2 K(M)[[t]];
n0
and extended to virtual bundles using the exponential rule
Symt(W1 W2) = Symt(W1) Symt(W2):
The effect on homotopy groups of the the oe-orientation therefore sends an al*
*most complex manifold M
of dimension 2n to
0 1
O
(ss*oeKTate)(M) = f!@ Sym qn(TC)A2 "K[[q]]0(S2n):
n1
This is often written as
0 1 0 1
O O du n
f!@ Sym qn(TC)A= Td@ M; Sym qn(TC)A-__
n1 n1 u
or simply as
0 1 0 1
O O
f!@ Sym qn(TC)A= Td@ M; Sym qn(TC)A:
n1 n1
The oe-orientation of KTatedetermines an invariant of Spin-manifolds, by insi*
*sting that the diagram
MSU ----! MU
?? ?
y ?y
MSpin ----! KTate
commute. To explain this invariant in classical terms, let M be a spin manifold*
* of dimension 2n, and, by
the splitting principle, write
TM ~=L1+ . .+.Ln
Q
for complex line bundles Li. The Spin structure gives a square root of Li, bu*
*t it is conventional to regard
each Lias having square root.
24 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
Since, for each i, the O(2) bundles underlying L1=2iand L-1=2iare isomorphic,*
* we can write
X -1=2 1=2
TM ~= Li+ Li - Li ;
which is a sum of SU-bundles.
Using this, one easily checks that the oe-orientation of M gives
0 1
O du n
Ab@M; Sym qn(TC)A -__ ;
n1 u
where the bAgenus is the push-forward in KO-theory associated to the unique ori*
*entation MSpin ! KO
making the diagram
MSU ----! MU ----! K
?? fl
y flfl
MSpin ----! KO ----! K
commute. As above, it is customary to suppress the grading and write
0 1
O
bA@M; Sym qn(TC)A;
n0
which is formula (27) in [Wit87].
We have proved
Proposition 2.67.The invariant
ss*MSpin ! Z[[q]]
associated to the oe-orientation on KTateis the Witten genus. *
* |___|
2.8. Modularity.
Proposition 2.68.For any element [M] 2 ss2nMU<6>, the series
-n
(ss2noeKTate)(M) -du_u 2 ss0KTate= Z[[q]]
is the q-expansion of a modular form.
Proof.Let us write
-n
(M) = (ss2noeKTate)(M) -du_u :
The discussion in the preceding section shows that (M) defines holomorphic func*
*tion on D, with integral
q-expansion coefficients. It suffices to show that, if ss :H ! D is the map
ss(o) = e2ssio;
then ss*(M) transforms correctly under the action of SL2Z. This follows from th*
*e discussion_of H in the
introduction. *
* |__|
3.Calculation of C_k(bGa; Gm )
In this section, we calculate the structure of the schemes C_k(bGa; Gm ) for *
*1 k 3, so as to be able to
compare them to BU<2k>HP in x4.
ELLIPTIC SPECTRA *
* 25
3.1. The cases k = 0 and k = 1. The group C_0(bGa; Gm )(R) is just the group of*
* invertible formal power
series f 2 R[[x]]; and C_1(bGa; Gm ) is the group of formal power series f 2 R[*
*[x]] with f(0) = 1. Let
R0= Z[b0; b-10; b1; b2; : :]:, and let R1= Z[b1; b2; b3; : :]:. If Fk 2 C_k(bGa*
*; Gm )(Rk) are the power series
X
F0= bixi
i0
X
F1= 1 + bixi;
i1
then the following is obvious.
Proposition 3.1.For k = 0 and k = 1, the ring Rk represents the functor C_k(bGa*
*; Gm_), with universal
element Fk. *
* |__|v
Note that F0 has a unique product expansion
Y
F0= a0 (1 - anxn) (3*
*.2)
n1
The aigive a different polynomial basis for R0 and R1.
3.2. The strategy for k = 2 and k = 3. For k 2, the group C_k(bGa; Gm )(R) is *
*the group of symmetric
formal power series f 2 R[[x1; : :;:xk]] such that f(x1; : :;:xk-1; 0) = 1 and
f(x1; x2; : :):f(x0+ x1; : :):-1f(x0; x1+ x2; : :):f(x0; x1; : :*
*):-1= 1:
In the light of Remark 2.12, we can replace the normalization f(x1; : :;:xk-1; *
*0) = 1 by f(0;Q: :;:0) = 1.
Alternatively, by symmetry, we can replace it by the condition that f(x1; : :;:*
*xk) = 1 (mod jxj).
Similarly, the group C_k(bGa; bGa)(R) is the group of symmetric formal power *
*series f 2 R[[x1; : :;:xk]] such
that f(x1; : :;:xk-1; 0) = 0 and
f(x1; x2; : :):- f(x0+ x1; : :):+ f(x0; x1+ x2; : :):- f(x0; x1; *
*: :):= 0:
We write C_kd(bGa; bGa)(R) for the subgroup consisting of polynomials of homoge*
*neous degree d.
Our strategy for constructing the universal 2 and 3-cocycles is based on the *
*following simple observation.
Lemma 3.3. Suppose that h 2 C_k(bGa; Gm )(R), and that h = 1 mod(x1; : :;:xk)d.*
* Then there is a
unique cocycle c 2 C_kd(bGa; bGa) such that h = 1 + c mod(x1; : :;:xk)d+1. If *
*g and h are two elements
of C_k(bGa; Gm ) of the form 1 + c mod(x1; : :;:xk)d+1, then g=h is an element *
*of C_k(bGa;_Gm ) of the form
1 mod(x1; : :;:xk)d+1. *
* |__|
We call c the leading term of h. We first calculate a basis of homogeneous po*
*lynomials for the group
of additive cocycles. Then we attempt construct multiplicative cocycles with st*
*udy with our homogeneous
additive cocycles as leading term. The universal multiplicative cocycle is the *
*product of these multiplicative
cocycles. Much of the work in the case k = 3 is showing how additive cocycles *
*can occur as leading
multiplicative cocycles.
In the cases k = 0 and k = 1, this procedure leads to the product description*
* (3.2)of invertible power
series.
We shall use the notation
ffix :C_k-1(bGa; Gm ) ! C_k(bGa; Gm )
for the map given in Definition 2.20, and reserve ffi for the map
ffi :C_k-1(bGa; bGa) ! C_k(bGa; bGa):
Definition 2.20 gives these maps for k 2; for f 2 C_0(bGa; Gm )(R) we define
ffixf(x1) = f(0)f(x1)-1
and similarly for C_0(bGa; bGa).
26 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
3.3. The case k = 2. Although we shall see (Proposition 3.12) that the ring OC_*
*2(bGa;Gm)is polynomial over
Z, the universal 2-cocycle F2 does not have a product decomposition
Y
F2= g2(d; bd);
d2
with g2(d; bd) having leading term of degree d, until one localizes at a prime *
*p. The analogous result for
H*BSU is due to Adams [Ada76].
Fix a prime p. For d 2, let c(d) 2 Z[x1; x2] be the polynomial
(1
_(xd+ xd- (x1+ x2)d)d = ps for somes 1
c(d) = p 1 2 (3*
*.4)
xd1+ xd2- (x1+ x2) otherwise
The following calculation of C_2(bGa; bGa) is due to Lazard; it is known as t*
*he "symmetric 2-cocycle lemma".
A proof may be found in [Ada74].
Lemma 3.5. Let A be a Z(p)-algebra. For d 2, the group C_2d(bGa; bGa)(A) is th*
*e free A-module_on the
single generator c(d). *
* |__|
Let
0 1
X tpk
E(t) = exp@ ___kA (3*
*.6)
k0 p
be the Artin-Hasse exponential (see for example [Haz78]). It is of the form 1 m*
*od(t), and it has coefficients
in Z(p).
For d 2, let g2(d; b) 2 C_2(bGa; Gm )(Q[b]) be the power series
( 2 d -p
g2(d; b) = ffix(E(bx ) i)fd is a powerpof (3*
*.7)
ffi2x(E(bxd))otherwise:
Using the formulae for the polynomials c(d) and the Artin-Hasse exponential, it*
* is not hard to check that
g2(d; b) belongs to the ring Z(p)[b][[x1; x2]], and that it is of the form
g2(d; b) = 1 + bc(d) mod(x1; x2)d+1: (3*
*.8)
We give the proof as Corollary 3.22.
Now let R2 be the ring
R2= Z(p)[a2; a3; : :]:;
and F22 C_2(bGa; Gm )(R2) be the the cocycle
Y
F2= g2(d; ad):
d2
Proposition 3.9.The ring R2 represents C_2(bGa; Gm ) x spec(Z(p)), with univers*
*al element F2.
Proof.Let A be a Z(p)-algebra, and let h 2 C_2(bGa; Gm )(A) be a cocycle. By Le*
*mma 3.5 and the equation
(3.8), there is a unique element a22 A such that
___h___= 1 mod(x ; x )3
g2(2; a2) 1 2
in C_2(bGa; Gm )(A). Proceeding by induction yields a unique homomorphism from *
*R2 to_A,_which sends the
cocycle F2 to h. *
* |__|
ELLIPTIC SPECTRA *
* 27
3.4. The case k = 3: statement of results. The analysis of C_3(bGa; Gm ) is mor*
*e complicated than that
of of C_2(bGa; Gm ) for two reasons. First, the structure of C_3(bGa; bGa) is m*
*ore complicated; in addition, it
is a more delicate matter to prolong some of the additive cocycles c into multi*
*plicative ones of the form
1 + bc + : :.:This is reflected in the answer: although the ring representing C*
*_2(bGa; Gm ) is polynomial, the
ring representing C_3(bGa; Gm ) x spec(Z(p)) contains divided polynomial genera*
*tors.
Definition 3.10.We write D[x] for the divided-power algebra on x over Z. It has*
* a basis consisting of the
elements x[m]for m 0; the product is given by the formula
x[m]x[n]= (m_+_n)!_m!n!x[m+n]:
If R is a ring then we write DR[x] for the ring R D[x].
We summarize some well-known facts about divided-power algebras in x3.4.1.
Fix a prime p. Let R3 be the ring
O
R3= Z(p)[ad|d 3 not of the form1 + pt] DZ(p)[a1+pt]:
t1
In x3.6.1, we construct an element F32 C_3(bGa; Gm ). In Proposition 3.28, we s*
*how that the map classifying
F3 gives an isomorphism
Z3= specR3-!~C_3(bGa; Gm ) x spec(Z(p)): (3.*
*11)
=
The plan of the rest of this section is as follows. In x3.5, we describe the *
*scheme C_k(bGa; bGa). We calculate
C_k(bGa; bGa) x specQ for all k, and we calculate C_3(bGa; bGa) x specFp. The p*
*roofs of the main results are
given in Appendix A.
In x3.6, we construct multiplicative cocycles with our additive cocycles as l*
*eading terms. This will allow
us to write a cocycle Z3 over R3 in x3.6.1. For some of our additive cocycles i*
*n characteristic p (precisely,
those we call c0(d)), we are only able to write down a multiplicative cocycle o*
*f the form 1+ac0(d) by assuming
that ap = 0 (mod p); these correspond to the divided-power generators in R3.
In x3.7, we show that the condition ap = 0 (mod p) is universal, completing t*
*he proof of the isomorphism
(3.11).
3.4.1. Divided powers. For convenience we recall some facts about divided-power*
* rings.
i.A divided power sequence in a ring R is a sequence
(1 = a[0]; a = a[1]; a[2]; a[3]; : :):
such that
a[m]a[n]= (m_+_n)!_m!n!a[m+n]
for all m; n 0. It follows that am = m!a[m]. We write D1(R) for the set of*
* divided power sequences in
R. It is clear that D1= specD[x].
ii.An exponential series over R is a series ff(x) 2 R[[x]] such that ff(0) = 1*
* and ff(x + y) = ff(x)ff(y). We
write Exp(R) for the set of such series. ItPis a functor from rings to abel*
*ian groups.
iii.Given a_2 D1(R), we define exp(a_)(x) = m0 a[m]xm 2 R[[x]]. By a mild ab*
*use, we allow ourselves to
write exp(ax) for this series. It is an exponential series, and the corresp*
*ondence a_7! exp(a_)(x) gives an
isomorphism of functors D1~=Exp. In particular both are group schemes.
iv.The map Q[x] ! DQ[x] sending x to x has inverse x[m]7! xm =m!, and this giv*
*es an isomorphism
D1x spec(Q) ~=A1x spec(Q):
v. We write Tp[x] for the truncated polynomial ring Tp[x] = Fp[x]=xp, and we w*
*rite ffp = specTp[x]. Thus
ffp(R) is empty unless R is an Fp-algebra, and in that case ffp(R) = {a 2 R*
* | ap = 0}.
28 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
P p-1
vi.Given a Z(p)-algebra R and an element a 2 R, we define texp(ax) = k=0akxk*
*=k!. Here we can divide
by k! because it is coprime to p.
vii.Over Fpthe divided power ring decomposes as a tensor product of truncated p*
*olynomial rings
O r
DFp[x] ~= Tp[x[p]]
r0
Moreover there is an equation
Y r r
exp(ax) = texp(a[px]p) (mod p):
r0
Each factor on the right is separately exponential: if a 2 ffp(R) then
texp(a(x + y)) = texp(ax) texp(ay):
In other words, the map
2]
a_7! (a[1]; a[p]; a[p; : :):
gives an isomorphism
Y
D1x spec(Fp) = ffp;
m0
and the resulting isomorphism
Y
ffp ~=Expx spec(Fp)
m0
is given by
Y m
b_7! texp(bm xp ):
m0
3.4.2. Grading. It will be important to know that the maps OC_k(bGa;Gm)! Rk we *
*construct may be viewed
as maps of connected graded rings of finite type: a graded ring R* is said to b*
*e of finite type over Z if each
Rn is a finitely generated abelian group.
We let Gm act on the scheme C_k(bGa; Gm ) by
(u:h)(x1; : :;:xk) = h(ux1; : :;:uxk);
*
* Q ff
and give OC_k(bGa;Gm)the grading associatedPto this action. One checks that the*
* coefficient of xff= ixi iin
the universal cocycle has degree |ff| = iffi. If k > 0 then the constant term*
* is 1 and the other coefficients
have strictly positive degrees tending to infinity, so the homogeneous componen*
*ts of OC_k(bGa;Gm)have finite
type over Z.
The divided power ring D[x] can be made into a graded ring by setting |x[m]| *
*= m|x|. We can then grade
our rings Rk by setting the degree of ad to be d. It is clear that R1 is a conn*
*ected graded ring of finite type
over Z, and Rk is a connected graded ring of finite type over Z(p)for k > 1.
This can be described in terms of an action of Gm on Zk = specRk. We have
Y
Z0~=Gm x A1
d1
Y
Z1~= A1
d1
Y
Z2~= A1x specZ(p)
d2
Y
Z3~= Z3;d
d3
ELLIPTIC SPECTRA *
* 29
where (
1x specZ d 6= 1 + pt
Z3;d= A (p)
D1x specZ(p)d = 1 + pt:
We let Gm act on A1 or Gm by u:a = ua, and on D1 by (u:a)[k]= uka[k]. We then l*
*et Gm act on Zk by
u:(ak; ak+1; : :):= (uk:ak; uk+1:ak+1; : :)::
The resulting grading on Rk is as described. For k 2, it is easy to check that*
* the map Zk ! C_k(bGa; Gm )
classifying Fk is Gm -equivariant.
As an example of the utility of the gradings, we have the following.
Proposition 3.12.The ring OC2(bGa;Gm)is polynomial over Z on countably many hom*
*ogeneous generators.
Proof.As OC2(bGa;Gm)is a connected graded ring of finite type over Z, it suffic*
*es by well-known arguments
to check that Z(p) OC2(bGa;Gm)is polynomial on homogeneous generators for all p*
*rimes p. By Proposition
3.9, we have an isomorphism of rings Z(p) OC2(bGa;Gm)~=OZ2 = Z(p)[ad | d 2], a*
*nd it is easy to check
that ad is homogeneous of degree d. *
* |___|
3.5. Additive cocycles. In this section we describe the group C_k(bGa; bGa)(A) *
*for various k and A. The
results provide the list of candidates for leading terms of multiplicative cocy*
*cles. Proofs are given in the
appendix A.
Fix an integer k 1. We write Ck(A) for C_k(bGa; bGa)(A), and we write Ckd(A)*
* for the subgroup C_kd(bGa; bGa)
of series which are homogeneous of degree d. Note that Ckd(A) = 0 for d < k.
P
Given a set I {1; : :;:k} we write xI = i2Ixi. One can easily check that f*
*or g 2 A[[x]] = C0(A) we
have
X
(ffikg)(x1; : :;:xk) = (-1)|I|g(xI):
I
For example, if g(x) = xd then
(ffi2g)(x;=y)(x + y)d- xd- yd
(ffi3g)(x;=y;-z)(x + y + z)d+ (x + y)d+ (x + z)d+ (y + z)d- xd- yd- *
*zd:
3.5.1. The rational case. Rationally, the cocycles ffikxd for d k are a basis *
*for the additive cocycles.
Proposition 3.13 (A.1).If A is a Q-algebra, then for d k the group Ckd(A) is t*
*he free abelian group on
the single generator ffikxd.
3.5.2. Divisibility. Now we fix an integer k 2 and a prime p.
Definition 3.14.For all n let p(n) denote the p-adic valuation of n. For d k w*
*e let u(d) be the greatest
common divisor of the coefficients of the polynomial ffik(xd). We write v(d) fo*
*r the p-adic valuation p(u(d)).
Let c(d) be the polynomial c(d) = ((-ffi)k(xd))=pv(d)2 Z[x1; : :;:xk] (We have *
*put a sign in the definition
to ensure that c(d) has positive coefficients). It is clear that
c(d) 2 Ckd(Z):
If we wish to emphasize the dependence on k, we write uk(d), ck(d), and vk(d).
We will need to understand the integers v(d) more explicitly.
Definition 3.15.For any nonnegative integer d and any prime p, we write oep(d) *
*for the sum of the digits
inPthe base p expansion of d. In morePdetail, there is a unique sequence of int*
*egers diwith 0 di< p and
idipi= d, and we write oep(d) = idi.
The necessary information is given by the following result, which will be pro*
*ved in Appendix A.
30 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
Proposition 3.16 (A.10).For any d k we have
ss
v(d) = max 0; k_-_oep(d)_p:- 1
The important examples of Proposition 3.16 for the present paper are k = 2 an*
*d k = 3:
Corollary 3.17.
(
v2(d)= 1 oep(d) = 1
0 otherwise
8
>>>2oe2(d) = 1 andp = 2
<1 oe (d) = 1 andp > 2
v3(d)= > p
>>:1oep(d) = 2
0 oep(d) > 2:
In other words, v2(d) = 1 if d is a power of p, and 0 otherwise. We have v3(d) *
*= 2 if p = 2 and d has the
form 2twith t > 1, and v3(d) = 1 if p = 2 and d has the form 2s(1 + 2t). On the*
* other hand, when p > 2 we
have v3(d) = 1 if d has the form ptor 2ptor ps(1 + pt) (with s 0 and t > 0). I*
*n all_other cases we have
v3(d) = 0. *
* |__|
In particular, the calculation of v2(d) shows that the cocycle c2(d) in Defin*
*ition 3.14 coincides with the
cocycle c(d) in the formula (3.4).
3.5.3. The modular case. We continue to fix an integer k 2 and a prime p, and *
*we analyze Ck(A) when
p = 0 in A.
For any ring A we define an endomorphism OE of A[[x1; : :;:xk]] by OE(xi) = x*
*pi. If p = 0 in A one checks
that this sends Ck(A) to Ck(A) and Ckd(A) to Ckdp(A). Moreover, if A = Fpthen a*
*p = a for all a 2 Fpand
thus OE(h) = hp.
In particular, we can consider the element OEjc(d) 2 Z[x1; : :;:xk], whose re*
*duction mod p lies in Ckpjd(Fp).
The following proposition shows that this rarely gives anything new.
Proposition 3.18.If p(d) v(d) then
j j (d)-v(d)+1
c(pjd) = c(d)p = OE c(d) (mod pp ):
It is clear from Proposition 3.16 that v(pd) = v(d), so even if the above pro*
*position does not apply to d,
it does apply to pid for large i.
Proof.We can reduce easily to the case j = 1. Write v = v(pd) = v(d), so that c*
*(d) = (-ffi)k(xd)=pv and
c(pd) = (-ffi)k(xpd)=pv. Write w = vp(d), so the claim is equivalent to the ass*
*ertion that
OE(-ffi)k(xd) = (-ffi)k(xpd) (mod pw+1):
P P
The left hand side is IOE(xdI) = IOE(xI)d. It is well-known that OE(xI) = (*
*xI)p (mod p), and that
whenever we have a = b (mod p) we also have api= bpi(mod pi+1). It followsPeasi*
*ly that OE(xI)d = (xI)pd __
(mod pw+1). As the right hand side of the displayed equation is just I(xI)pd,*
* the claim follows. |__|
3.5.4. The case k = 3. In this section we set k = 3, and we give basis for the *
*group of additive three-cocycles
over an Fp-algebra. In order to describe the combinatorics of the situation, it*
* will be convenient to use the
following terminology.
Definition 3.19.We say that an integer d 3 has type
I if d is of the form 1 + ptwith t > 0.
II if d is of the form ps(1 + pt) with s; t > 0.
IIIotherwise.
ELLIPTIC SPECTRA *
* 31
If d = ps(1 + pt) has type I or II we define c0(d) = OEsc(1 + pt) 2 C3d(Fp). No*
*te that d has type I precisely
when oep(d - 1) = 1, and in that case we have c0(d) = c(d).
Proposition 3.20 (A.12).If A is an Fp-algebra then C3(A) is a free module over *
*A generated by the
elements c(d) for d 3 and the elements c0(d) for d of type II.
3.6. Multiplicative cocycles. We fix a prime p and an integer k 1. In this sec*
*tion we write down the
basic multiplicative cocycles. We need the following integrality lemma; many si*
*milar results are known (such
as [Haz78, Lemma 2.3.3]) and this one may well also be in the literature but we*
* have not found it.
Lemma 3.21. Let A be a torsion-free p-local ring, and OE:A -!A a ring map such *
*that OE(a) = ap (mod p)
for all a 2PA. If (bk)k>0 is a sequence of elements such that OE(bk) = bk+1 (mo*
*d pk+1) for all k, then the
series exp( kbkxpk=pk) 2 (Q A)[[x]] actually lies in A[[x]].
P k
Proof.Write f(x)Q= exp( kbkxp =pk). Clearly f(0) = 1, so there are unique elem*
*ents aj 2 Q A such
that f(x) = j>0E(ajxj), and it is enough to show that aj2 A for all j. By taki*
*ng logs we find that
X k X pi i
bkxp =pk = aj xjp=pi:
k i;j
P pj
It follows that aj = 0 unless j is a power of p, and that bk = k=i+jpiapi. We*
* may assume inductively
that a1; ap; : :;:apj-1are integral. It follows that for i < j we have OE(api) *
*= appi(mod p), and thus (by a
well-known lemma) that
j-i-1 pj-i-1 pj-i j-i
OE(appi ) = OE(api) = api (mod p ):
It follows that
j-1X j-i
pjapj= bj- piappi
i=0
j-1X j-i-1
= bj- piOE(api)p (mod pj)
i=0
= bj- OE(bj-1)
= 0 (mod pj);
or in other words that apjis integral. *
* |___|
Recall from (3.6)that E(t) 2 Z(p)[[t]] denotes the Artin-Hasse exponential.
Corollary 3.22.If d is such that p(d) v(d), then ffikxE(bxd)p-v(d)2 Q[b][[x1; *
*: :;:xk]] actually lies in
C_k(bGa; Gm )(Z(p)[b]) Z(p)[b][[x1; : :;:xk]]. It has leading term bc(d).
Proof.The symmetric cocycle conditions are clear, so we need only check that th*
*e series is integral. Using
the expin the Artin-Hasse exponential gives the formula
0 1 0 1
-v(d) X bpiffik(xdpi) X bpic(dpi)
ffikxE(bxd)p = exp@ _________i+v(d)A= exp@ _______iA:
i0 p i0 p
In view of Lemma 3.21, it suffices to check that OE(c(dpi)) = c(dpi+1) (mod pi+*
*1), where OE is the endomor-_
phism of Z(p)[[x1; : :;:xk]] given by OE(xi) = xpi. This follows from Propositi*
*on 3.18. |__|
Definition 3.23.If R is a Z(p)-algebra, b is an element of R, and if d is such *
*that p(d) v(d), we define
-v(d)
E(k; d; b) def=ffikxE(bxd)p
to be the element of C_k(bGa; Gm )(R) given by the corollary.
32 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
In order to analyze the map ffix, we need the following calculation.
Lemma 3.24. If p(d) v(d) we have
E(k; d; b)p = E(k; pd; bp) (mod p):
Proof.We can work in the universal case, where A = Z(p)[b] is torsion-free, so *
*it makes sense to use expo-
nentials. We have X
k k k
E(k; d; b) = exp( bp c(p d)=p );
k
and it follows easily that E(k; d; b)p=E(k; pd; ap) = exp(pac(d)). One checks e*
*asily that the series_exp(pt)-1
has coefficients in pZ(p), and the claim follows. *
* |__|
We need one other family of cocycles, given by the following result.
Proposition 3.25.Let B be the divided-power algebra on one generator b over Z(p*
*). Then the series
ffikxexp(bxd=pv(d)) = exp((-1)kb c(d)) lies in C_k(bGa; Gm )(B) B[[x1; : :;:xk*
*]]. |___|
3.6.1. The case k = 3. Suppose that d 3 is not of the form 1 + pt. Then Corol*
*lary 3.17 shows that
p(d) v(d), and so Definition 3.23 gives cocycles
g3(d; ad) def=E(3; d; ad) 2 C_3(bGa; Gm )(Z(p)[ad]): (3.*
*26)
For d = 1 + ptand t 1, let
g3(d; ad) def=exp(-adc(d)) 2 C_3(bGa; Gm )(DZ(p)[ad])
be the cocycle given by Proposition 3.25.
Note that if d = 1 + ptthen in Fp DZ(p)[ad] we have an equation
Y [ps]
g3(d; ad) = texp(-ad c0(dps)) (3.*
*27)
s0
*
* s]
as in x3.4.1, and each factor on the right is separately an element of C_3(bGa;*
* Gm )(Tp[a[pd]).
Let F3 be the cocycle
Y
F3= g3(d; ad)
d3
over O
Z3= specZ(p)[ad | d 6= 1 + pt] DZ(p)[a1+pt]:
t1
Proposition 3.28.The map Z3! C_3(bGa; Gm ) x spec(Z(p)) classifying F3 is an is*
*omorphism.
Proof.Let h denote this map. It is easy to check that it is compatible with the*
* Gm -actions described in
x3.4.2, so the induced map of rings preserves the gradings.
We will show that the map h(R): Z3(R) -!C_3(bGa; Gm )(R) is an isomorphism wh*
*en R is a Q-algebra
or an Fp-algebra. This means that the map h*:OC_3(bGa;Gm) Z(p)-!OZ3 becomes an *
*isomorphism after
tensoring with Q or Fp. As both sides are connected graded rings of finite type*
* over Z(p), it follows that h
is itself an isomorphism.
Suppose that R is a Q-algebra. In this case we get divided powers for free, a*
*nd an element of Z3(R)
is just a list of elements (a3; a4; : :):. According to Proposition 3.13, the *
*additive cocycle c(d) generates
C_3d(bGa; bGa)(R). Since gd has leading term adc(d), the process of successive*
* approximation suggested by
Lemma 3.3 shows that h(R) is an isomorphism.
We now suppose instead that R is an Fp-algebra. As DFp[x] = Tp[x[pi]| i 0], *
*we see that a point of
*
* i]
Z3(R) is just a sequence of elements ad 2 R for d 3, with additional elements *
*ad;i= a[pdwhen d has type
ELLIPTIC SPECTRA *
* 33
I, such that ad;0= ad and apd;i= 0. We write a0dpi= ad;i. With this reindexing,*
* an element of Z3(R) is a
system of elements ad (where d has type II or III) together with a system of el*
*ements a0d(where d has type
I or II) subject only to the condition (a0d)p = 0.
On the other hand, suppose that f 2 C3(bGa; Gm )(R) is a cocycle with leading*
* term c of degree d. If d has
type III, then Proposition 3.20 shows that c = adc(d) for a unique c in R. If d*
* has type I, then c = a0d;0c0(d)
for some unique a0d;0in R. Finally, if d has type II, then c = adc(d) + a0dc0(d*
*) for some unique ad and a0din
R. We shall show in Proposition 3.29 that in fact (a0d)p = 0. The process of su*
*ccessive approximation_gives
a point of Z3(R) which clearly maps to f under the map h(R). *
* |__|
In the course of the proof, we used the following result, whose proof will be*
* given in x3.7.
Proposition 3.29.Suppose that R is an Fp-algebra and that f 2 C3(bGa; Gm )(R) h*
*as leading term ac0(d)
(so that d has type I or II). Then ap = 0.
Corollary 3.30.The ring OC3(bGa;Gm)is a graded free abelian group of finite typ*
*e.
Proof.Proposition 3.28 shows that this is true p-locally for every prime p, so *
*it is true integrally. |___|
3.7. The Weil pairing: cokernel of ffix :C_2(bGa; Gm ) ! C_3(bGa; Gm ). The fir*
*st result of this section is a
proof of Proposition 3.29, which completes the calculation in Proposition 3.28.*
* The analysis which leads to
this result also gives a description of the cokernel of the map
ffix3
C_2(bGa; Gm ) -! C_(bGa; Gm );
which we shall use to compare C_3(bGa; Gm ) to BU<6>HP .
More precisely, the scheme Z3 decomposes as a product of schemes
Z3= Z03x Z003
where
Z03= specDZ(p)[a1+pt| t 1]
Z003= specZ(p)[ad | d not of the form1 + pt]:
We shall show that ffix maps C_2(bGa; Gm ) x specFpsurjectively onto Z003x spec*
*Fp, and that the cokernel
Z03x specFphas a natural description as the scheme Weil(bGa) of Weil pairings. *
*In x4.5.1, we shall see that
this scheme is isomorphic to the scheme associated to the even homology of K(Z;*
* 3). In this paper we give
a bare-bones account of Weil pairings. The reader can consult [Bre83, Mum65, AS*
*98] for a more complete
treatment.
Definition 3.31.Let R be any ring, and h an element of C_3(bGa; Gm )(R). We def*
*ine a series e(h) 2 R[[x; y]]
by the formula
p-1Yh(x; kx; y)
e(h)(x; y) = ________:
k=1h(x; ky; y)
In x3.7.1, e will be interpreted as giving a map of group schemes
C_3(bGa; Gm ) x specFp-!Weil(bGa):
Proposition 3.32.We have
e(h)(x; y) e(h)(x;=z)e(h)(x; y + z) h(px;_y;_z)h(x; py; p*
*z)
= e(h)(x; y + z) (mod p);
and e(h)(x; y)p = 1 (mod p).
34 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
Proof.Recall the cocycle relation R(w; x; y; z) = 1, where
R(w; x; y; z) = h(x;_y;_z)h(w;_x_+hy;(z)w:+ x; y; z)h*
*(w; x; z)
By brutally expanding the relation
R(y; z; k(y + z); x)R(ky; (k + 1)z; y; x)R((k + 1)z; y; ky; x)
R(ky; kz; z; x)R(kx; x; y; z)R(x; *
*y; z; kx) = 1;
and using the symmetry of h, we find that
h(x;_kx;_y):h(x;_kx;_z)= __h(x;_kx;_y_+_z):_____h(x;_ky;_kz)_:h((k_+_1)*
*x;_y; z)
h(x; ky; y) h(x; kz; z) h(x; ky + kz; y + z)h(x; (k + 1)y; (k +h1)z)(k*
*x; y; z)
We now take the product from k = 1 to p - 1. We note that the second term on th*
*e right has the form
f(k)=f(k + 1), so the product gives f(1)=f(p). After dealing with the last term*
* in a similar way and doing
some cancellation, we find that
e(h)(x; y) e(h)(x; z) = e(h)(x; y + z) h(px; y; z) h(x; py; p*
*z)-1;
as claimed. For any cocycle h we have h = 1 (mod xyz), so our expression reduce*
*s to e(h)(x; y + z) modulo
p. This means that e(h) behaves exponentially in the second argument, so e(h)(x*
*; y)p_= e(h)(x; py) = 1
(mod p). *
* |__|
We can also consider an additive analogue of the above construction. Given c *
*2 C3(bGa; bGa)(R), we write
p-1X
e+(c)(x; y) = (c(x; kx; y) - c(x; ky; y)):
k=1
By applying the definitions and canceling in a simple-minded manner we find that
e+(ffi3f)(x; y) = f(x) - f(x + py) - f(y) + f(y + px) - f(px) + f*
*(py):
Thus e+(ffi3f) = 0 (mod p).
The following calculation is the key to the proof of Proposition 3.29, and it*
* also permits the identification
of Z03with the scheme of Weil pairings.
Lemma 3.33. Let d = ps(1 + pt) with s 0 and t 1. Then
s ps+t ps+tps
e+(c0(d)) = xp y - x y (mod p):
Proof.As c0(ps(1 + pt))p = c(1 + pt)ps+1, it suffices to calculate e+(c(1 + pt)*
*) (mod p). Let n = 1 + pt. By
Corollary 3.17, we have c(n) = ffi3(xn)=p, so that
pe+(c(n))= xn - (x + py)n - yn + (y + px)n - pnxn + pnyn
= -pnxn-1y + pnxyn-1 (mod p2)
t pt 2
= p(xyp - x y) (mod p ):
Thus e+(c(1 + pt)) = xypt- yxpt(mod p) as required. *
* |___|
We can now give the
Proof of PropositionS3.29.uppose that R is an Fp-algebra and that h 2 C3(bGa; G*
*m )(R) has leading term
ac0(d) (so that d has type I or II).
It is easy to see that e(h) = 1 + ae+(c0(d)) (mod (x; y; z)d+1), and thus tha*
*t e(h)p = 1 + ape+(c0(d))p
(mod (x; y; z)pd+1). On the other hand, we know from Proposition 3.32 that e(h)*
*p = 1. Lemma 3.33 shows __
that e+(c0(d))pis a nonzero polynomial over Fpwhich is homogeneous of degree pd*
*. It follows that ap = 0. |__|
ELLIPTIC SPECTRA *
* 35
3.7.1. The scheme of Weil pairings. In this section we work implicitly over spe*
*c(Fp). We note that a faithfully
flat map of schemes is an epimorphism.
We also recall [DG70 , III,x3,n. 7] that the category of affine commutative g*
*roup schemes over Fpis an
abelian category, in which specf :specA ! specB is an epimorphism if and only i*
*f f :B ! A is injective.
Let R be an Fp-algebra. We write Weil(bGa)(R) for the group (under multiplica*
*tion) of formal power series
f(x; y) 2 R[[x; y]] such that
f(x; x)= 1
f(x; y)f(x;=z)f(x; y + z) (3.*
*34)
f(x; z)f(y;=z)f(x + y; z):
Note that this implies f(x; y)f(y; x) = 1 by a polarization argument. We write *
*Weil(bGa)(R) = ; if R is not
an Fp-algebra.
Proposition 3.32 shows that, if R is an Fp-algebra and h 2 C_3(bGa; Gm ) is a*
* three-cocycle, then e(h) is a
Weil pairing. In other words, e may be viewed as a natural transformation
e:C_3(bGa; Gm ) -!Weil(bGa):
In this section, we show that there is a commutative diagram
ffix e
C_2(bGa; Gm_)//_C_3(bGa;_Gm/)///_Weil(bGa)OOOO (3.*
*35)
| ~=| ~=|
fflfflfflffl|||| ||
Z003//______//Z03x Z003___////_Z03
of group schemes over specFp, with exact rows and with epi, mono, and isomorphi*
*sms as indicated. In x4.5,
we compare the top row to a sequence arising from the fibration K(Z; 3) ! BU<6>*
* ! BSU.
To begin, we note that Weil(bGa) is an affine group scheme over Fp. The repre*
*senting ring OWeil(bGa)is the
quotient of the ring Fp[akl| k; l 0] by the ideal generated by the coefficient*
*s of the series "f(x; x) - 1 and
f"(x + y; z) - "f(x; z)f"(y; z) and "f(x; y + z) - "f(x; y)f"(x; z), where "fis*
* the power series
f"(x; y) = X aklxkyl:
We let Gm act on Weil(bGa) by (u:f)(x; y) = f(ux; uy), and this gives a grading*
* on OWeil(bGa)making it into
a graded connected Hopf algebra over Fp. If
X
f(x; y) = aijxiyj
i;j
is the universal Weil pairing, then the degree of aijis i + j.
*
* P
Lemma 3.36. The ring OWeil(bGa)is a tensor product of rings of the form Fp[a]=a*
*p. If f = aijxiyj is the
universal Weil pairing, then elements of the form apm;pnwith m < n are a basis *
*for Ind(OWeil(bGa)):
Proof.Let us temporarily write A for OWeil(bGa). Note that if f(x; y) 2 Weil(bG*
*a)(R) we have f(x; y)p =
f(x; py) = f(x; 0) = 1, and it follows that the Frobenius map for A is trivial.*
* It follows from the structure
theory of connected graded Hopf algebras over Fpthat A is a tensor product of r*
*ings of the form Fp[a]=ap.
The dual of the group of indecomposables in A is easily identified with the k*
*ernel of the map
Weil(bGa)(Fp[ffl]=ffl2) -!Weil(bGa)(Fp)
that is induced by the augmentation map Fp[ffl]=ffl2 -!Fp. This kernel is the s*
*et of power series of the form
1 + fflg(x; y) (with g 2 Fp[[x; y]]) such that 1 + fflg(x; x) = 0 and (1 + fflg*
*(x; y))(1 + fflg(x; z)) = 1 + fflg(x; y + z)
and (1 + fflg(x; z))(1 + fflg(y; z)) = 1 + fflg(x + y; z). This reduces to the *
*requirementPthat g(x;my)nbe additive in
both arguments, with g(x; x) = 0. The additivity means that g(x; y) must have t*
*he form m;nbmnxp yp .
Because g(x; x) = 0 we must have bmm = 0 (even if p = 2) and bmn = -bnm if m > *
*n. |___|
36 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
Let j denote the splitting map
Z03! C_3(bGa; Gm ):
Note that Z03is a group scheme, because
Y Y
Z03~= D1x spec(Z(p)) ~= Expx spec(Z(p)):
It is easy to check that j is a map of group schemes (even over spec(Z(p))). Th*
*e first step in the analysis of
the diagram 3.35 is the following.
Proposition 3.37.The map of group schemes
ej :Z03-!Weil(bGa)
is an isomorphism.
Q
Proof.First, when R is an Fp-algebra we can identify Z03(R) with d{a 2 R | ap *
*= 0}, where d runs over
integers d 3 of type I or II, and according to (3.27), j(a_) is the cocycle
Y
j(a_) = texp(-adc0(d)):
d
Lemma 3.33 shows that if d = ps(1 + pt) then
s ps+t ps+tps d+1
e(texp(-adc0(d)) = 1 - ad(xp y - x y ) (mod (x; y) ):
It follows that ej induces an isomorphism of indecomposables. Moreover, ej indu*
*ces a map of graded rings
if OZ03is given the grading with a0din dimension d. We thus a map of connected *
*graded algebras, both of
which are tensor products of polynomial algebras truncated at height p, and our*
* map gives an isomorphism_
on indecomposables. It follows that the map is an isomorphism. *
* |__|
To show that Z003is the kernel of e, we first observe that C_2(bGa; Gm ) maps*
* to the kernel.
Lemma 3.38. If R is an Fp-algebra and g 2 C_2(bGa; Gm )(R) then e(ffixg) = 1.
Proof.By definition we have ffixg(x; kx; y) = g(x; y)g(kx; y)=g((k + 1)x; y). A*
*s ffixg is symmetric, we have
ffixg(x; ky; y) = g(x; y)g(x; ky)=g(x; (k + 1)y). By substituting these equatio*
*ns into the definition of e(ffixg)_
and canceling, we obtain e(ffixg)(x; y) = g(x; py)=g(px; y), which is 1 because*
* p = 0 in R. |__|
Next we show that ffix actually factors through the inclusion Z003! C_3(bGa; *
*Gm ). Let w and be given by
the formulae
w(2)= 1
w(d)= v3(d) - v2(d) d 3
(d)= pw(d)d:
By Corollary 3.17, it is equivalent to set w(d) = 1 if d is of the form ps(1 + *
*pr) with r 0, and w(d) = 0
otherwise. It follows also that gives a bijection from {d | d 2} to {d | d 3*
* andd is not of the1form+
pt}.
Let r: Z2= specR2! Z003be given by the formula
w(d)
r*a(d)= apd :
It is clear that r is faithfully flat.
Lemma 3.39. The diagram
ffix 3
C_2(bGa; Gm-)---!C_(bGa; Gm )
x x
~=?? ??
Z2 --r--! Z003
ELLIPTIC SPECTRA *
* 37
commutes over spec(Fp). In particular, over spec(Fp), ffix factors through a fa*
*ithfully flat map C_2(bGa; Gm ) !
Z003:
Proof.This follows from the equations
-v2(()d) pw(d) pw(d) *
* pw(d)
ffixg2(d; a) = ffi3xE(axd)p = E(3; d; a) = E(3; (d); a ) = g*
*3((d); a ):
The only equation which is not a tautology is the third, which is Lemma 3.24. A*
*ctually the lemma does not
apply in the case d = 2, but the result is valid anyway. One can see this direc*
*tly from the_definitions, using
the fact that ffi3(x2) = 0. *
* |__|
Proposition 3.40.The kernel of the map
e:C_3(bGa; Gm ) -!Weil(bGa)
is Z003(which is thus a subgroup scheme). Moreover, we have C_3(bGa; Gm ) = Z03*
*x Z003as group schemes.
Proof.We know from Lemma 3.38 that effix = 1, and faithfully flat maps are epim*
*orphisms of schemes, so
Lemma 3.39 implies that Z00 ker(e). As the map (f0; f00) 7! f0f00gives an isomo*
*rphism Z0x Z00-!C_3,
and e: Z0-! Weil(bGa) is an isomorphism, it follows that Z00= ker(e). This mean*
*s that Z00is a subgroup
scheme, and we have already observed before Proposition 3.37 that the same is t*
*rue of_Z0._It follows that
C_3= Z0x Z00as group schemes. *
* |__|
We summarize the discussion in this section as the following.
Corollary 3.41.If we work over spec(Fp) then the following sequence of group sc*
*hemes is exact:
C_2(bGa; Gm ) ffi-!C_3(bGa; Gm ) e-!Weil(bGa) ! 0:
*
*|___|
3.8. The map ffix :C_1(bGa; Gm ) ! C_2(bGa; Gm ). In the course of comparing BS*
*UHP to C_2(bGa; Gm ) in x4,
we shall use the following analogue of Corollary 3.41.
Proposition 3.42.For each prime p, the map
ffix2
C_1(bGa; Gm ) x spec(Fp) -! C_(bGa; Gm ) x spec(Fp)
is faithfully flat.
Proof.In order to calculate ffix, it is useful to use the model for C_1which is*
* analogous to our model Z2 for
C_2: Let Z1 be the scheme
Z1= specZ(p)[ad | d 1];
and let
Y
F1def= E(1; d; ad)
d1
Y
= E(adxd)
d1
be the resulting cocycle over OZ1. It is clear that the map
Z1! C_1(bGa; Gm ) x spec(Z(p))
classifying F1 is an isomorphism. Thus if k = 1 or 2, and if R is a Z(p)-algebr*
*a, then Zk(R) is the set of
sequences (ak; ak+1; : :):of elements of R.
For d 1 let (d) = pv2(d)d, with the convention that v2(1) = 1. The calculati*
*on of v2(d) in Corollary
3.17 shows that induces a bijection from the set {d | d 1} to the set {d | d *
* 2}. Let r: Z1! Z2 be the
map which sends a sequence a_= (a1; a2; : :):2 Z1(R) to the sequence
v2(d)
r(a_)(d)= apd :
38 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
Thus r is a product of copies of the identity map A1! A1 (indexed by {d | v(d) *
*= 0}), together with some
copies of the Frobenius map A1 ! A1 (indexed by {d | v(d) = 1}). These maps are*
* faithfully flat, and so r
is faithfully flat. The Proposition then follows once we know that the diagram
ffix 2
C_1(bGa; Gm ) x spec(Fp)----!C_(bGa; Gm ) x spec(Fp)
x x
~=?? ??~=
Z1x spec(Fp) --r--! Z2x spec(Fp)
commutes. The commutativity of the diagram follows from the equations (modulo p)
v2(d)
ffixE(axd)= E(2; d; a)p
v2(d)
= E(2; (d); ap )
v2(d)
= g2((d); ap ):
The first and last equations are tautologies; the middle equation follows from *
*Lemma 3.24. |___|
3.9. Rational multiplicative cocycles. Given k > 0, let Yk(R) be the set of for*
*mal power series f(x) 2
R[[x]] such that f(x) = 1 (mod xk). This clearly defines a closed subscheme Yk *
* C_0(bGa; Gm )
Proposition 3.43.Over spec(Q), the map ffikx:Yk -!C_k(bGa; Gm ) is an isomorphi*
*sm.
Proof.Let R be a Q-algebra, and let g 2 R[[x1; : :;:xk]] be an element of C_k(b*
*Ga; Gm )(R). We need to
show that g = ffikx(f)Pfor a unique element f 2 Yk(R). If I = (x1; : :;:xk) th*
*en g = 1 (mod I) so the
series log(g) = - m>0(1 - g)m =m is I-adically convergent. One checks that it*
* defines an element of
C_k(bGa; Ga)(R), so Proposition 3.13 tellsPus that there is a unique h 2 R[[x]]*
* with h = 0 (mod xk) and
ffik(h) = log(g). The series exp(h) = mhm =m! is x-adically convergent to an *
*element of_Yk(R),_which is
easily seen to be the required f. *
* |__|
4.Topological calculations
In this section we will compare our algebraic calculations with known topolog*
*ical calculations of E*BU,
H*BSU, and H*BU<6>, and we deduce that BUE = C_k(PE; Gm ) for k 3. We start*
* with the cases
k = 0 and k = 1, which are merely translations of very well-known results. We t*
*hen prove the result for all
k when E = HPQ (the rational periodic Eilenberg-MacLane spectrum); this is an e*
*asy calculation.
Next, we prove the case k = 2 with E = HP. It suffices to do this with coeffi*
*cients in the field Fp, and
then it is easy to compare our analysis of the scheme C_2(bGa; Gm ) to the shor*
*t exact sequence
PHP -! BUHP -! BSUHP :
For BU<6> we recall Singer's calculation of H*(BU<6>; Fp), which is based on *
*the fibration
K(Z; 3) -!BU<6> -!BSU:
Most of the work in this section is to produce the topological analogue of the *
*exact sequence
ffix e
C_2(bGa; Gm_)//_C_3(bGa;_Gm/)/_Weil(bGa)//_0
of Corollary 3.41; see (4.9). Having done so, we can easily prove the isomorphi*
*sm BU<6>E ~=C_3(PE; Gm ) for
E = HPFp. The isomorphism for integral homology follows from the cases E = HPQ *
*and E = HPFp. Using
a collapsing Atiyah-Hirzebruch spectral sequence and its algebraic analogue, we*
* deduce the case E = MP,
and we find that MP0BU<6> is free over MP0. It is then easy to deduce the isomo*
*rphism for arbitrary E.
ELLIPTIC SPECTRA *
* 39
4.1. Ordinary cohomology. We begin with a brief recollection of the ordinary co*
*homology of BU, in
order to fix notation.
It is well-known that H*BU is a formal power series algebra generated by the *
*Chern classes. It follows
easily that the corresponding thing is true for HP0BU: we can define Chern clas*
*ses ck 2 HP0BU for
k > 0 and wePfind that HP0BU = Z[[ck | k > 0]]. We also put c0 = 1. We define a*
*Pseries c(t) 2 HP0[[t]]
by c(t) = k0 cktk. We then define elements qk by the equation tc0(t)=c(t) = *
* kqktk. The group of
primitives is
X Y
PrimHP0BU = { niqi| ni2 Z} ~= Z:
i i>0
There is an inclusion S1 = U(1) j-!U and a determinant map U det--!S1 with de*
*tOj = 1. These give maps
P -Bj-!BU B-det--!P with B detOBj = 1, and the fiber of B detis BU<4> = BSU. In*
* fact, if i:BSU -!BU
is the inclusion then one sees easily that i+j :BSU xP -!BU induces an isomorph*
*ism of homotopy groups,
so it is an equivalence.
We have HP0P = Z[[x]] with B det*x = c1 and Bj*c1 = x and Bj*ck = 0 for k > 1*
*. It follows (as is
well-known) that the inclusion BSU -!BU gives an isomorphism HP0BSU = HP0BU=c1=*
* Z[[ck | k > 1]].
In particular, both BU and BSU are even spaces.
The Hopf algebra HP0BU is again a polynomial algebra, with generators bk for *
*k > 0. We also put
b0= 1. The pairing between this ring and HP0BU satisfies
( Q
Y 1 if bffi= bk
= i i 1
i 0 otherwise.
The group of primitives in HP0BU is generated by elements rk, which are charact*
*erized by the equation
X
t d log(b(t))=dt = t b0(t)=b(t) = rktk:
k
4.2. The isomorphism for BU<0> and BU<2>.
Proposition 4.1.For k = 0 and k = 1 and for any even periodic ring spectrum E, *
*the natural map
BU<2k>E -!C_k(PE; Gm )
is an isomorphism.
Proof.We treat the case k = 1, leaving the case k = 0 for the reader. A coordi*
*nate x on PE gives
isomorphisms
OPE = E0P ~=E0[[x]]
O_PE= eE0P~=E0{fi1; fi2; : :}:
E0(BU) ~=E0[b1; b2; : :]:
OC1(PE;Gm)~=E0[b01; b02; : :]::
Here the fii 2 eE0P are defined so = ffiij, and bi = (E0ae1)(fii), wh*
*ere ae1:P -! BU classifiesPthe
virtual bundle 1 - L. The b0iare defined by writing the universal element of C_*
*1(PE; Gm ) as 1 + i1 b0ixi.
By Definition 2.27, the map BUE ! C_1(PE; Gm ) classifies the elementPb 2 E0B*
*U bE0P ~=E0BU[[x]] __
which is the adjoint of the map E0ae1. It is easy to see that b = ibixi. *
* |__|
Recall that Cartier duality (2.2)gives an isomorphism
PE ~=Hom_(PE; Gm ):
The construction f 7! 1=f gives a map
Hom_(PE; Gm ) i-!C_1(PE; Gm ):
40 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
Corollary 4.2.The diagram
(B det)E
PE ___________//BUE
~=|| |~=|
fflffl|i fflffl|
Hom_(PE; Gm_)__//_C_1(PE; Gm )
commutes.
Proof.It will be enough to show that the dual diagram of rings commutes. As E0B*
*U is generated over E0
by (E0ae1)(Ee0P), it suffices to check commutativity after composing with E0ae1*
*. It is then clear,_because
B detOae1 classifies det(1 - L) ~=L-1, and so has degree -1. *
* |__|
4.3. The isomorphism for rational homology and all k.
Proposition 4.3.For any k > 0 we have
HP0(BU<2k>; Q) = HP0(BU; Q)=(c1; : :;:ck-1) = Q[[cn | n k]]:
We also have an isomorphism
BU<2k>HPQ ~=C_k(bGa; Gm ) x spec(Q):
Proof.We have fibrations BU<2k + 2> -!BU<2k> -!K(Z; 2k). It is well-known that
H*(K(Z; 2k); Q) = Q[u2k]
with |u2k| = 2k. We know that the map BU<2k> -!K(Z; 2k) induces an isomorphism *
*on ss2k(-) and we
may assume inductively that H*(BU<2k>; Q) = Q[[cn | n k]], so the Hurewicz the*
*orem tells us that u2k
hits a nontrivial multiple of ck. It now follows from the Serre spectral sequen*
*ce that
H*(BU<2k + 2>; Q) = Q[[cn | n k + 1]] = H*(BU; Q)=(c1; : :;:ck):
Dually, we know that H*(BU; Q) is generated by primitive elements risuch that r*
*iis dual to ci, and we find
that H*(BU<2k>; Q) = Q[ri| i k]. These are precisely the functions on C1(bGa; *
*Gm ) that are unchanged
when we replace f 2 C1(bGa; Gm )(R) by f exp(g) for some polynomial g of degree*
* less than k, as we see from
the definition of the ri. We see from the proof of Proposition 3.43 that these *
*are the same as the functions
that depend only on ffik-1x(f), and thus that BU<2k>HPQ can be identified with *
*C_k(bGa;_Gm ) x spec(Q), as
claimed. *
* |__|
4.4. The ordinary homology of BSU.
Proposition 4.4.The natural map
BSUHP -! C_2(bGa; Gm )
is an isomorphism.
Proof.It is enough to prove this modulo p for all primes p, so fix one. Consid*
*er the diagram of affine
commutative group schemes (in which everything is taken implicitly over Fp)
PHP __________//BUHP_______//_BSUHP
~=|| ~=|| ||
fflffl| fflffl|ffix fflffl|
Hom_(bGa; Gm_)_//_C_1(bGa;_Gm/)/_C_2(bGa; Gm ):
The diagram commutes by Corollaries 2.28 and 4.2. The splitting BU = BSU x P im*
*plies that the top row
is a short exact sequence. It is clear that Hom_(bGa; Gm ) is the kernel of ffi*
*x, so it remains to_show that ffix is
an epimorphism. That is precisely the content of Proposition 3.42. *
* |__|
ELLIPTIC SPECTRA *
* 41
4.5. The ordinary homology of BU<6>. The mod p cohomology of BU<2k> was compute*
*d (for all k 0)
by Singer [Sin68]. We next recall the calculation for k = 3. Note that BU<6> *
*is the fiber of a map
BSU -!K(Z; 4) and K(Z; 4) = K(Z; 3) so we have a fibration
K(Z; 3) fl-!BU<6> v-!BSU:
In this section we give an algebraic model for the ordinary homology of this fi*
*bration, in terms of the theory
of symmetric cocycles and Weil pairings.
Classical calculations show that for p > 2 we have
H*(K(Z; 3); Fp) = E[u0; u1; : :]: Fp[fiu1; fiu2; : :]:;
where |uk| = 2pk+ 1 and uk+1= Ppkuk and fiu0= 0. WeQwrite A* for the polynomial*
* subalgebra generated
by the elements fiuk for k > 0. We also write A = k0A2k, which is an ungraded*
* formal power series
algebra over Fp. In the case p = 2 we have
H*(K(Z; 3); F2) = F2[u0; u1; : :]:;
with |uk| = 2k+1+ 1 and uk+1= Sq2k+1uk, and we let A* be the subalgebra generat*
*ed by the elements u2k.
We write A_*for the vector space dual Hom(A*; Fp).
Lemma 4.5. In the Serre spectral sequence
H*(BSU; H*(K(Z; 3); Fp)) =) H*(BU<6>; Fp)
the class utsurvives to E2pt+2, and then there is a differential d2pt+2(ut) = q*
*1+pt, up to a unit in Fp.
Proof.We treat the case p > 2 and leave the (small) modifications for p = 2 to *
*the reader. As BU<6> is
5-connected, we must have a transgressive differential d4(u0) = c2 (up to a uni*
*tPin Fp). We can think of
H*(BU; Fp) as a ring of symmetric functions in the usual way, so we have c2 = *
* i 0. We also have q1 = c1 (which vanishes on BSU) and thus Ppt-1: :P:pP1*
*(c2) = -q1+ptin
H*(BSU; Fp). It follows from the Kudo transgression theorem and our knowledge *
*of the action of_the
Steenrod algebra that utsurvives to E2pt+2and d2pt+2(ut) = q1+pt. *
* |__|
Proposition 4.6.We have a short exact sequence of Hopf algebras
H*(BSU; Fp)=(c2; q1+pt| t > 0) ae H*(BU<6>; Fp) i A*:
Moreover, H*(BU<6>; Fp) is a polynomial ring over Fp, concentrated in even degr*
*ees, with the same Poincare
series as Fp[ck | k 3].
Proof.Note that qk = kck modulo decomposables, so we can take q1+ptas a generat*
*or of H2(1+pt)(BSU)
p-locally when t > 0. Thus
H*(BSU; Fp) = Fp[q1+pt| t > 0] Fp[ck | k 2 is not of the1form+ p*
*t]
Using this, one can check that Lemma 4.5 gives all the differentials in the spe*
*ctral sequence, and that
E1 = H*(BU; Fp)=(c2; q1+pt| t > 0) A*
= Fp[ck | k 2 is not of the1form+ pt]
Fp[fiuk | k > 0]:
By thinking about the edge homomorphisms of the spectral sequence, we obtain th*
*e claimed short exact
sequence of Hopf algebras. As the two outer terms are polynomial rings in even *
*degrees, the same is true_of
the middle term. As |fiuk| = |q1+pk|, we have the claimed equality of Poincare *
*series. |__|
Corollary 4.7.BU<6> is a even space, and H*BU<6> is a polynomial algebra of fin*
*ite type over Z.
42 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
Proof.It is easy to see that BU<6> has finite type. The remaining statements ar*
*e true p-locally for all_p_by
the Proposition, and the integral statement follows because everything has fini*
*te type. |__|
Corollary 4.8.The sequence of group schemes over Fp
BSUHPFp -!BU<6>HPFp -!spec(A_) -!0
is exact.
4.5.1. The Weil scheme and HP0K(Z; 3). In this section, we work over Fpunless o*
*therwise specified. In
particular, homology is taken with coefficients in Fp.
We now have the solid arrows of the diagram
BSUHP ______//_BU<6>HP___//spec(A_)O_//_0 (4*
*.9)
f2|~=| f3|| ~=OO
fflffl|ffix fflffl|e fflffl
C_2(bGa; Gm_)//_C_3(bGa;_Gm/)/_Weil(bGa)//_0:
The diagram commutes by Corollary 2.28. Moreover the rows are exact (by Corolla*
*ries 3.41 and 4.8), and
the map f2 is an isomorphism by Proposition 4.4. It follows that there is a map*
* making the diagram
commute. Our next task is to show that this map is an isomorphism.
We can give an explicit formula for this map. Recall from x2.3.3 that f3 clas*
*sifies the 3-cocycle ^ae32
HP0P bHP0BU<6>. Here ^ae3is the adjoint of HP0ae3, where ae3 is the map
P3 ae3-!BU<6>
Q
whose composite to BU classifies the bundle i(1 - Li). Let W :P2 ! BU<6> be th*
*e map whose composite
to BU classifies the virtual bundle
p-1X *
* p-1X
(1 - L1)(1 - Lk1)(1 - L2) - (1 - L1)(1 - Lk2)(1 - L2) ~=(1 - L1)(1 -(L*
*2)4.1Lk2-0Lk1:)
k=1 *
* k=1
Let ^Wbe the adjoint in HP0P2b HP0BU<6> of the map HP0W. Let x = -c1L1 and y *
*= -c1L2 be the
indicated generators of HP0P2. Then ^Wgives a power series
W^(x; y) 2 HP0(BU<6>)[[x; y]] ~=HP0P2b HP0BU<6>:
Lemma 4.11. The power series ^W(x; y) has coefficients in the subring A_[[x; y]*
*]. As such it is an element
of Weil(bGa)(A_). The map : spec(A_) ! Weil(bGa) classifying ^W(x; y) makes the*
* diagram (4.9)commute.
Proof.Recall that the map e:C_3(bGa; Gm ) ! Weil(bGa) takes the power series f(*
*x; y; z) to the power series
p-1Yf(x; kx; y)
e(f)(x; y) = ________:
k=1f(x; ky; y)
Recall also that the H-space structure of BU<6> corresponds on the algebraic si*
*de to the multiplication of
power series and on the topological side to addition of line bundles. The H-spa*
*ce structure of P corresponds
on the algebraic side to addition in the group bGaand on the topological side t*
*o the tensor product of line
bundles.
Putting these observations together shows that
W^= e(^ae3):
The lemma follows from this equation and the structure of the solid diagram (4.*
*9). |___|
Lemma 4.12. For s 1, we have an equation
s ps 2
W*q1+ps= p(xyp - x y) mod p
in the integral cohomology HP0P2.
ELLIPTIC SPECTRA *
* 43
Proof.As x = -c1L1 and y = -c1L2, the total Chern class of the bundle (4.10)is *
*given by the formula
W*c(t) = (1_-_yt)(1_-_pxt)(1_-_(x_+(py)t)1:- xt)(1 - py*
*t)(1 - (px + y)t)
We have q(t) = td logc(t). Modulo p2 we have equations
td log(1 - xt)-=tx(1 + xt + (xt)2+ : :):
td log(1 - pxt)-=pxt
td log(1 - (x + py)t)(=x_+_py)t_(1 - (x + py)t)
= - pyt(1 + xt + (xt)2+ : :):- xt(1 + xt + (xt)2+ : *
*:):
- pxyt2(1 + 2xt + 3(xt)2+ : :)::
With these formulae it is easy to verify the assertion. *
* |___|
Note that Lemma 4.11 implies that the map (of Fp-modules)
HP0W :HP0P2 -!HP0BU<6>
factors through the inclusion of A_ in HP0BU<6>.
Proposition 4.13.The map of group schemes : spec(A_) ! Weil(bGa) is an isomorph*
*ism.
Proof.First note that A is a formal power series algebra on primitive generator*
*s (because u0 is primitive
and the Steenrod action preserves primitives). It follows that A_ is a divided *
*power algebra over Fpand
thus a tensor product of rings of the form Fp[y]=yp. We know from Lemma 3.36 th*
*at OWeil(bGa)also has
this structure. It will thus suffice to show that the map Ind(OWeil(bGa)) -! I*
*nd(A_) = Prim(A)_ is an
isomorphism, or equivalently that the resulting pairing of Ind(OWeil(bGa)) with*
* Prim(A) is perfect.
Define elements bi2 H*P by setting = ffiij, and define elements bij2*
* A_ by setting
bij= H*W(bi bj):
It is clear that the Weil pairing g(x; y) associated to HP0W is given by the fo*
*rmula
X
g(x; y) = bijxiyj:
ij
P
Let f(x; y) = i;jaijxiyj be the universal Weil pairing defined over OWeil(b*
*Ga), so our map sends f to
g and thus aijto bij. We know from Lemma 3.36 that the elementsmapi;pj(with i <*
* j) form a basis for
Ind(OWeil(bGa)). On the other hand, the elements (fiuk)p (with k > 0 and m 0)*
* are easily seen to form a
basis for Prim(A).
The calculation of the Serre spectral sequence in Lemma 4.5 and the character*
*istic class calculation in
Lemma 4.12 together imply that
k pk
W*fiuk = ffl(xyp - x y)
in H*(P2), where ffl is a unit in Fp. It follows that the inner product in A is the same (up to
a unit) as the inner product in H*(P2), and this i*
*nner product is just_ffiimffijk.
This proves that the pairing is perfect, as required. *
* |__|
Corollary 4.14.For periodic integral homology, the map BU<6>HP -!C_3(bGa; Gm ) *
*is an isomorphism.
Proof.It is enough to prove this mod p for all p. We can chase the diagram 4.9*
* to see that the map
BU<6>HPFp -!C_3(bGa; Gm ) x spec(Fp) is an epimorphism. We see from Proposition*
*s 4.3 and 3.28 that the __
corresponding graded rings have the same Poincare series, so the map must actua*
*lly be an isomorphism. |__|
44 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
4.6. BSU and BU<6> for general E. Let FGL be the scheme of formal group laws an*
*d let G = bA1xFGL.
There is a canonical group structure oe :G xFGL G = bA2x FGL ! bA1x FGL = G giv*
*en by the formula
oe(a; b; F) = (a +F b; F). We define an actionPof Gm on FGL by (u:F)(x; y) = u-*
*1F(ux; uy). This gives
a grading on OFGL; explicitly, if F(x; y) = i;jaijxiyj is the universal forma*
*l group law, then aijis a
homogeneous element of OFGL of degree i + j - 1. It is clear that OFGL is gener*
*ated (subject to many
relations) by the elements aij. It is a theorem of Lazard (see [Ada74] for exam*
*ple) that OFGL is a graded
polynomial algebra with one generator in each degree i > 0.
The scheme C = C_3(G; Gm ) is the functor that assigns to each ring R the set*
* of pairs (F; f), where F is
a formal group law over R and f 2 R[[x1; x2; x3]] is symmetric, congruent to 1 *
*modulo x1x2x3, and satisfies
the cocycle condition
f(x1; x2; x3)f(x0+F x1; x2; x3)-1f(x0; x1+F x2; x3)f(x0; x1; x3)-*
*1 = 1:
The action of Gm on FGL extends to an action on C by the formula u:(F; f) = (u:*
*F; u:f), where
(u:f)(x1; x2;=x3)f(ux1; ux2; ux3)
and
(u:F)(x; y)= u-1F(ux; uy):
P *
* j
This gives OC the structure of a graded OFGL-algebra. If f(x1; x2; x3) = i;j;*
*k0bijkxi1x2xk3then bijkcan
be thought of as a homogeneous element of OC with degree i + j + k. Moreover, w*
*e have b000= 1.
It is clear that OC is generated over OFGL by the elements bijk, and thus tha*
*t OC is a connected graded
ring of finite type over Z.
Lemma 4.15. The ring OC is a graded free module over OFGL. In particular, it is*
* free of finite type over
Z.
Proof.Let I be the ideal in OC generated by the elements of positive degree in *
*OFGL, so the associated
closed subscheme V (I) ~=spec(Z) FGL just consists of the additive formal grou*
*p law. It follows that
OC=I = OC_3(bGa;Gm), which is a free abelian group by Corollary 3.30. We choose*
* a homogeneous basis for
OC=I and lift the elements to get a system of homogeneous elements in OC. Using*
* these, we can construct
a graded free module M over OFGL and a map M -ff!OC of OFGL-modules that induce*
*s an isomorphism
M=IM ~=OC=IOC. It is easy to check by induction on the degrees that ff is surje*
*ctive. Also, M is free
over OFGL, which is free over Z, so M is free over Z. Now, if we have a surject*
*ive map f :A -!B of finitely
generated Abelian groups such that A is free and A Q ~=B Q, it is easy to see*
* that f is an isomorphism.
Thus, if we can show that M has the same rational Poincare series as OC, we can*
* deduce that ff is an
isomorphism.
If (F; f) is a point of C over a rational ring R, then we can define a series*
* expFin the usual way and
get a series g = f O (exp3F) defined by g(x1; x2; x3) = f(expF(x1); expF(x2); e*
*xpF(x3)). Clearly we have g 2
C_3(bGa; Gm )(R), and this construction gives an isomorphism C x spec(Q) -!FGL *
*xC_3(bGa; Gm ) x spec(Q).
It follows that the Poincare series of OC is the same as that of OFGL OC_3(bGa*
*;Gm), which is the same as
that of M by construction. *
* |___|
Proposition 4.16.For any even periodic ring spectrum E, the natural maps
BSUE -!C_2(PE; Gm )
and
BU<6>E -!C_3(PE; Gm )
are isomorphisms.
Proof.Let k = 2 or 3. Because BU<2k> is even, we know that the Atiyah-Hirzebruc*
*h spectral sequence
H*(BU<2k>; E*) =) E*BU<2k>
ELLIPTIC SPECTRA *
* 45
collapses, and thus that E1BU<2k> = 0 and E0BU<2k> is a free module over E0. If*
* we have a ring map
E0 -!E between even periodic ring spectra then we get a map E0 E00E00BU<2k> -! *
*E0BU<2k>, and
a comparison0of Atiyah-Hirzebruch spectral sequences shows that this is an isom*
*orphism, so BU<2k>E =
BU<2k>E xSE0SE. On the other hand, because the formation of C_kcommutes with ba*
*se change, we have
C_k(PE; Gm ) = C_k(PE0xSE0SE; Gm ) = C_k(PE0; Gm ) xSE0SE:
It follows that if the theorem holds for E0then it holds for E. It holds for E *
*= HP by Proposition 4.4 or
Corollary 4.14, and we have ring maps
HP -!HPQ -!HPQ ^ MU = MPQ;
so the theorem holds for MPQ.
For any E, we can choose a coordinate on E and thus a map MP -!E of even peri*
*odic ring spectra, so
it suffices to prove the theorem when E = MP, in which case SE = FGL. In this c*
*ase we have a map of
graded rings OC -!MP0BU<2k> = MU*BU<2k>, both of which are free of finite type *
*over Z. This map is a
rational isomorphism by the previous paragraph, so it must be injective, and th*
*e source and target must have
the same Poincare series. It will thus suffice to prove that it is surjective. *
*Recall that I denotes the kernel of
the map MP0-! Z = HP0 that classifies the additive formal group law, or equival*
*ently the ideal generated
by elements of strictly positive dimension in MU*. By induction on degrees, it *
*will suffice to prove that the
map OC=I -!MP0BU<2k>=I is surjective. Base change and the Atiyah-Hirzebruch seq*
*uence identifies this
map with the map OC_3(bGa;Gm)-!HP0BU<2k>, in other words the case E = HP of the*
* proposition. This
case was proved in Proposition 4.4 (k = 2) or Corollary 4.14 (k = 3). *
* |___|
Appendix A.Additive cocycles
The main results of this section are proofs of Propositions 3.13, 3.16, and 3*
*.20. We use the notation of
x3. In particular, we abbreviate Ck(A) for C_k(bGa; bGa), and for d 1 we write*
* Ckd(A) for the subgroup of
polynomials which are homogeneous of degree d.
For d 1 let xd be considered as an element of C0(bGa; bGa)(Z). Then we have *
*polynomials ffik(xd) 2
Z[x1; : :;:xk] giving elements of Ck(Z). For example
ffi2(xd)= xd1+ xd2- (x1+ x2)d
ffi3(xd)= xd1+ xd2+ xd3- (x1+ x2)d- (x1+ x2)d- (x2+ x3)d+ (x1+ x2+ x3*
*)d:
A.1. Rational additive cocycles.
Proposition A.1 (3.13).If A is a Q-algebra, then for d k the group Ckd(A) is t*
*he free abelian group on
the single generator ffikxd.
Proof.If h 2 Ck(A) then there is a unique series f(x) such that h(x; ffl; : :;:*
*ffl) = fflk-1f(x) (mod fflk), and
moreover f(0) = 0. It follows that there is a unique series g 2 C0k(A) whose (k*
* - 1)'st derivative is f. We
can thus define an A-linear map ss :Ck(A) -!C0k(A) by ss(h) = (-1)kg. We claim *
*that this is the inverse
of ffik.
To see this, suppose that g 2 C0k(A), so that g(k-1)(0) = 0. From the definit*
*ions, we have
X
(ffikg)(x; ffl;=: :;:ffl)(-1)|I|(g(|I|ffl) - g(x + |I|ffl))
I
k-1X
= (-1)j k - 1 (g(jffl) - g(x + jffl));
j=0 j
46 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
where I runs over subsets of {2; : :;:k}. To understand this, we introduce the *
*operators (Tf)(x) = f(x + ffl)
and (Df)(x) = f0(x). Taylor's theorem tells us that T = exp(fflD). It is clear *
*that
k-1X
(-1)j k - 1 g(x + jffl)= ((1 - T)k-1g)(x)
j=0 j
= ((1 - exp(fflD))k-1g)(x)
= (-ffl)k-1g(k-1)(x) (mod fflk):
If we feed this twice into our earlier expression and use the fact that g(k-1)(*
*0) = 0, we find that
(ffikg)(x; ffl; : :;:ffl) = (-1)kfflk-1g(k-1)(x) (mod ffl*
*k):
This shows that ssffik = 1.
To complete the proof, it suffices to show that ss is injective. Suppose that*
* h 2 Ck(A) and that ss(h) = 0,
so that h(x; ffl; : :;:ffl) = 0 (mod fflk). If k = 2 we consider the cocycle co*
*ndition
h(y; z) - h(x + y; z) + h(x; y + z) - h(x; y) = 0:
If we substitute z = ffl and work modulo ffl2 then the first two terms become z*
*ero and we have h(x; y + ffl) =
h(x; y), or equivalently @h(x; y)=@y = 0. By symmetry we also have @h(x; y)=@x *
*= 0, and as A is rational
we can integrate so h is constant. We also know that h(0; 0) = 0 so h = 0 as re*
*quired.
Now suppose that k > 2. We know that h has the form g(x1; : :;:xk)xk for some*
* series g. By assumption,
fflk divides h(x; ffl; : :;:ffl) = fflg(x; ffl; : :;:ffl) so g(x; ffl; : :;:ffl*
*) = 0 (mod fflk-1). On the other hand, x2; : :;:xk-1
also divide g so it is not hard to see that g(x; ffl; : :;:ffl; 0) = g(x; ffl; *
*: :;:ffl) = 0 (mod fflk-1). Moreover, the
series g(x1; : :;:xk-1; 0) lies in Ck-1(A), so by induction on k we find that g*
*(x1; : :;:xk-1; 0) = 0. This__
shows that h(x1; : :;:xk-1; ffl) = 0 (mod ffl2). The argument of the k = 2 case*
* now shows that h = 0. |__|
A.2. Divisibility. Recall that ud is the greatest common divisor of the coeffic*
*ients of the polynomial ffikxd.
Let
kxd
c(k; d) = ffi_u:
d
It is clear that Ck(Z) = Ck(Q) \ Z[[x1; : :;:xk]], so Proposition A.1 has the f*
*ollowing corollary.
Corollary A.2.For d k, the group Ckd(Z) is a free abelian group on the single *
*generator c(k; d) |___|
We fix a prime p and an integer k 1. In x3 it is convenient work p-locally, *
*and then to use the cocycles
k(xd)
c(d) = (-ffi)__;
pv(d)
which locally at p are unit multiples of c(k; d) (see Definition 3.14). In this*
* section we study v(d) = p(u(d)).
It is clear that u(d) is the greatest common divisor of the multinomial coeff*
*icients
____d____;
a1! . . ...ak!
P
where ai 1 and ai= d.
We start with some auxiliary definitions.
Definition A.3.For any nonnegative integer d, we write oep(d) for the sum of th*
*e digits in thePbase p
expansion of d. In more detail, there is a unique sequence of integers diwith 0*
* di< p and idipi= d,
ELLIPTIC SPECTRA *
* 47
P
and we write oep(d) = idi. Given a sequence ff = (ff1; : :;:ffk) of nonnegati*
*ve integers, we write
X
|ff|= ffi
Y i
xff= xffii
Yi
ff!= ffi!
i
supp(ff)= {i | ffi> 0}:
Lemma A.4. We have
v(d) = inf{p(d!=ff!) | |ff| = d andffi> 0 for alli}: |___|
To exploit this, we need some well-known formulae involving multinomial coeff*
*icients.
Lemma A.5. We have vp(n!) = (n - oep(n))=(p - 1).
Proof.The number of integers in {1; : :;:n} that are divisible by p is bn=pc. *
*Of these, preciselyPbn=p2c
are divisible by a furtherPpower of p, and so on. ThisPleads easily to the for*
*mula vp(n!) = kbn=pkc.
IfPn has expansion inipi in base p, then bn=pkc = iknipi-k. A little manipu*
*lation gives vp(n!) =
*
* __
ini(pi- 1)=(p - 1) = (n - oep(n))=(p - 1) as claimed. *
* |__|
Corollary A.6.For any multi-index ff we have
!
X
vp(|ff|!=ff!) = oep(ffi) - oep(|ff|)=(p - 1):
i
Thus aeP fi oe
oep(ffi) - oep(d)fi __
v(d) = inf __i___________pf-i1fi|ff| = d andffi>:0 for|alli_|
It is not hard to check the following description of the minimum in Corollary*
* A.6.
Lemma A.7. The minimum in Corollary A.6 is achieved by the multi-index ff such*
* that summing
d = ff1+ . .+.ffk
in base p involves "carrying" the fewest number of times; and v(d) is equal to *
*the number of carries. |___|
The proof of Proposition 3.16 involves working out this number of carries. To*
* make the argument precise,
we introduce a few definitions.
Definition A.8.We let A(p; k; d) denote the set of doubly indexed sequences ff *
*= (ffij), where i runs from
1 to k, j runs over all nonnegative integers, and the following conditions are *
*satisfied:
i.For eachPi; j we have 0 ffij p - 1.
ii.We have i;jffijpj= d.
iii.For each i there exists j such that ffij> 0.
*
* P
By writing multi-indices in base p, we see that v(d) is the minimum value of *
*( ijffij- oep(d))=(p - 1) as
ff runs over A(p; k; d).
Definition A.9.We let B = B(p; k; d) be the set of sequences fi = (fij) (where *
*j runs over nonnegative
integers) such that
i.For eachPj we have 0 fij k(p - 1).
ii.We have Pjfijpj= d.
iii.We have jfij k.
48 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
We also write eB=PeB(p; k; d) for the larger set of sequences satisfying only c*
*onditionsPi and ii. Given fi 2 eB
we write o(fi) = jfij, so fi 2 B if and only if o(fi) k. If d has expansion *
*d = k"fikpk in base p, then
"fi= ("fi0; "fi1; : :):is an element of eB, with o("fi) = oep(d).
Proposition A.10 (3.16).For any d k we have
ss
v(d) = max 0; k_-_oep(d)_p:- 1
Alternatively, v(d) is equal to the minimum number of "carries" in base-p arith*
*metic, when d is calculated
as the sum of k integers a1; : :;:ak with ai 1.
P
Proof.Consider the map ae: A(p;Pk; d) -!B(p; k; d) defined by ae(ff)j = iffij*
*. It is easily seen that ae is
surjective and that oae(ff) = ijffij. It follows that v(p; k; d) = inf{(o(fi)*
* - oep(d))=(p - 1) | fi 2 B}. If
k oep(d) = o("fi) then "fi2 B and this makes it clear that v(d) = 0. From now *
*on we assume that k > oep(d).
P P
We define a map : eB\ B -!Beas follows. If fi 2 eB\ B then jfij< k and jf*
*ijpj= d. As d k this
clearly cannot happen unless there exists some i > 0 with fii> 0. We let j deno*
*te the largest such i. We
then define 8
>**i = j - 1 fij-1+ p
:i 6= j - 1;fjii:
We claim that the resulting sequence lies in eB. The only way this could fail w*
*ould be if fij-1+ p > k(p - 1),
but as fij> 0 this would imply
o(fi) fij+ fij-1 1 + (k - 1)(p - 1) k;
contradicting the assumption that fi 62 B.
Note that o(fi) = o(fi) + (p - 1). It follows that for some i, the sequence f*
*i = i("fi) is defined, lies in B,
and satisfies k o(fi) = oep(d) + i(p - 1) < k + p - 1. It follows that
ss
i = o(fi)_-_oep(d)_p=- 1k_-_oep(d)_p;- 1
and thus that v(d) d(k - oep(d))=(p - 1)e. By definition we have o(fl) k for *
*all fl 2 B, and this_implies
the reverse inequality. Thus v(d) = d(k - oep(d))=(p - 1)e. *
* |__|
A.3. Additive cocycles: The modular case. In this section we give the descripti*
*on of C3(A) when A is
an Fp-algebra, as promised in Proposition 3.20. For convenience, we recall what*
* we need to prove.
Let OE be the endomorphism of A[[x1; : :;:xk]] defined by OE(xi) = xpi, and w*
*e observed that if p = 0 in A
then this sends Ck(A) to Ck(A) and Ckd(A) to Ckdp(A). Moreover, if A = Fpthen a*
*p = a for all a 2 Fpand
thus OE(h) = hp.
Definition A.11.We say that an integer d 3 has type
I if d is of the form 1 + ptwith t > 0.
II if d is of the form ps(1 + pt) with s; t > 0.
IIIotherwise.
If d = ps(1 + pt) has type I or II we define c0(d) = OEsc(1 + pt) 2 C3d(Fp). No*
*te that d has type I precisely
when oep(d - 1) = 1, and in that case we have c0(d) = c(d).
Proposition A.12 (3.20).If A is an Fp-algebra then C3(A) is a free module over *
*A generated by the
elements c(d) for d 3 and the elements c0(d) for d of type II.
The proof will be given at the end of this section. ItPis based on the observ*
*ation that a cocycle h =
h(x; y; z) 2 C3d(A) can be written uniquely in the form ihi(x; y)zi. Each him*
*ust be a two-cocycle, and so
a multiple of c2(d - i). The symmetry of h restricts how the hican occur.
ELLIPTIC SPECTRA *
* 49
It is convenient to have the following description of the image of OE.
Lemma A.13. If p = 0 in A and h 2 Ck(A) and h(x1; : :;:xk-1; ffl) = 0 (mod ffl*
*2) then h = OE(g) for some
g 2 Ck(A). Moreover, if h is homogeneous of degree d, then g is homogeneous of *
*degree d=p, which means
that h = 0 if p does not divide d.
Proof.The cocycle condition gives
h(x1; : :;:xk) - h(x1; : :;:xk-1; xk+ ffl) + h(x1; : :;:xk-1+ xk; ffl) - *
*h(x1; : :;:xk-2; xk; ffl) = 0:
Modulo ffl2, the last two terms vanish and we conclude that @h=@xk = 0. This sh*
*ows that powers xjkcan
only occur in h if p divides j, or in other words that h is a function of xpk. *
*By symmetry it is a function of
xpifor all i, or in other words it has the form OE(g) for some g. It is easy to*
* check that g lies_again in Ck(A).
The extra statements for when h is homogeneous are clear. *
* |__|
Definition A.14.Given an integer d 3 and a prime p, we let o = o(d) be the uni*
*que integer such that
po+ 1 < d po+1+ 1.
Definition A.15.We define a map ss :C3d(A) -!A as follows. Given a cocycle h 2 *
*C3d(A), write
Xd
h(x; y; z) = hi(x; y)zi:
i=0
P d
Then we can write h(x; y; z) uniquely in the form i=0hi(x; y)zi. It is easy t*
*o check that hiis a two-cocycle,
and so Lemma 3.5 implies that hi= aic2(d - i) for a unique element ai2 A. Set s*
*s(h) = apo(d).
Lemma A.16. There is a unit 2 Fpx such that ss(ac(d)) = a, so ss is always su*
*rjective. If d is not
divisible by p then ss :C3d(A) -!A is an isomorphism. If d is divisible by p th*
*en the kernel of ss is contained
in the image of the map OE:C3d=p(A) -!C3d(A).
Proof.For the first claim we need only checkothat when A = Fp, the element = s*
*s(c(d)) is nonzero.
Equivalently, we claim that some term xiyjzp (with i + j + po = d) occurs nont*
*rivially in c(d). Given
Corollary A.6 and Proposition 3.16, it is enough to show that there exist integ*
*ers i; j > 0 with i + j + po = d
and ss
oep(i)_+_oep(j)_+_1_-=oep(d)max3_-_oep(d)_;:0
p - 1 p - 1
If oep(d) 3 then this reduces to the requirement that oep(i)+oep(j) = oep(d)*
*-1. We cannot have d = po+1
or d = po+1+ 1 becausePin those cases oep(d) < 3, so we must have po+ 1 < d < p*
*o+1. It follows that in the
base-p expansion d = oi=0dipiwe have do > 0, and thus that oep(d - po) = oep(*
*d) - 1 2. It is now easy
to find numbers i; j > 0 such that i + j = d - po and the sum can be computed i*
*n base p without carrying,
which implies that oep(i) + oep(j) = oep(d - po) as required.
We now suppose that oep(d) 2. In this case, we need to find i; j > 0 such th*
*at i + j + po = d and
3 - oep(d) oep(i) + oep(j) + 1 - oep(d) < 3 - oep(d) + p - *
*1;
or equivalently
2 oep(i) + oep(j) < p + 1:
Assuming that p > 2, the possible values of d, together with appropriate values*
* of i and j, are as follows.
d = po+1 i = j = 1_2(p - 1)po
d = 1 + po+1 i = 1 ; j = (p - 1)po
d = ps+ po (0 < s o)i = ps-1; j = (p - 1)ps-1
In the case p = 2, the possibilities are as follows.
d = 2o+1 (o > 0) i = j = 2o-1
d = 1 + 2o+1 i = 1 ; j = 2o
d = 2s+ 2o (0 < s < o)i = j = 2s-1
50 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
This completes the proof that = ss(c(d)) is nonzero. For general A we have ss(*
*ac(d)) = a, and it follows
immediately that ss is surjective.
We next show that the kernel of ss is contained in the image of OE (and thus *
*is zero if p does not divide
d). Suppose that h 2 C3d(A) and ss(h) = 0. Let aibe as in Definition A.15, so t*
*hat apo= ss(h) = 0. By
Lemma A.13, it suffices to check that h is divisible by x2. We already know tha*
*t it is divisible by x, so we
just need to know that a1= 0. Let i;j2 Fpbe the coefficient of xiyj in c2(()i +*
* j), so we have
X
h = i;jakxiyjzk:
i+j+k=d
As h is symmetric in x, y, and z, we conclude that i;jak = i;kaj. In particular*
*, we have
a1po;d-po-1= apo1;d-po-1= 0:
It is thus enough to check that po;d-po-1is a unit in Fp. In the case d = po+1+*
* 1 we have c(2; p; po+1) =
o+1
((x + y)po+1- xpo+1- ypo+1)=p and thus po;d-po-1= ppo =p. Corollary A.6 te*
*lls us that this
integer has p-adic valuation 0, so it becomes a unit in Fp. In the case when d*
* < po+1+ 1, we have
c(p; 2; d - 1) = (x + y)d-1- xd-1- yd-1and thus po;d-1-po= dp-o1 . It is not *
*hard to see that we
P o
have a base-p expansion d - 1 = i=0dipiin which do > 0. Given this, Corollary*
* A.6 again_tells us that
po;d-1-pois a unit, as required. *
* |__|
Lemma A.17. If d has type II then ss(c0(d)) = 0.
Proof.We have d = ps(1+pt) with s > 0 and 1+pt 3. As s > 0 we have 1+ps+t< ps+p*
*s+t 1+ps+t+1,
so o(ps+ ps+t) = s + t. We thus have to prove that there aretno terms of the fo*
*rm xiyjzps+tin c(1 + pt)ps,
or equivalently that there are no terms of the form xiyjzp in c(1 + pt). This i*
*s clear because c(1_+ pt) has
the form xyz f(x; y; z), where f is homogeneous of degree pt- 2. *
* |__|
Proof of Proposition A.12.It is clear from Lemma A.16 that C3(A) is generated o*
*ver A by the elements
OEsc(d) for all s and d. However, Proposition 3.18 and Corollary 3.17 tell us t*
*hat OEsc(d) = c(psd) unless
p(d) < v(d), where
8
>>>2oe2(d) = 1 andp = 2
<1 oe (d) = 1 andp > 2
v(d) = > p
>>:1oep(d) = 2
0 oep(d) > 2:
Suppose that d is one of these exceptional cases. We clearly cannot have oep(d)*
* > 2. If oep(d) = 1 then d = pt
for some t. The inequality p(d) < v(d) means that t < 2 if p = 2 and t < 1 if p*
* > 2. We also must have
d 3, so t > 0, and t > 1 if p = 2. These requirements are inconsistent, so we *
*cannot have oep(d) = 1.
This only leaves the possibility oep(d) = 2, so d = pr(1 + pt) with t 0, and t*
* > 0 if p = 2. The inequality
p(d) < v(d) now means that r = 0. The inequality d 3 means that the case t = 0*
* is excluded even when
p > 2.
In other words, OEsc(d) = c(dps) unless s > 0 and d has the form 1 + ptwith t*
* > 0, so psd has type II.
Thus C3(A) is spanned by the elements c(d) for d 3 and c0(d) for d of type II.
It is easy to see that C3d(A) = C3d(Fp) A, and in the case A = Fpwe know fro*
*m Lemmas A.16 and A.17_
that our spanning set is linearly independent. The proposition follows. *
* |__|
Appendix B. Generalized elliptic curves
In this appendix, we outline the theory of generalized elliptic curves. We ha*
*ve tried to give an elementary
account, with explicit formulae wherever possible. This has both advantages and*
* disadvantages over the other
available approaches, which make more use of the apparatus of schemes and sheaf*
* cohomology. For more
information, and proofs of results merely stated here, see [Del75, KM85, Sil94,*
* DR73]. Note, however, that
ELLIPTIC SPECTRA *
* 51
our definition is not quite equivalent to that of [DR73 ]: their generalized el*
*liptic curves are more generalized
than ours, so what we call a generalized elliptic curve is what they would call*
* a stable curve of genus 1 with
a specified section in the smooth locus.
We shall again think of non-affine schemes as functors from rings to sets. T*
*he basic example is the
projective scheme Pn, where Pn(R) is the set of submodulesPL Rn+1 such that L *
*is a summand and has
rank one. If we have elements a0; : :;:an 2 R such that iRai= R then the vect*
*or (a0; : :;:an) 2 Rn+1
generates such a submodule, which we denote by [a0 : : :::an]. This is of cours*
*e a free module. In general,
L may be a non-free projective module, so it need not have the form [a0 : : :::*
*an], but nonetheless it is
usually sufficient to consider only points of that form. For more details, and *
*a proof of equivalence with
more traditional approaches, see [Str99a, Section 3].
Definition B.1.A Weierstrass curve over a scheme S is a (non-affine) scheme of *
*the form
C= C(a1; a2; a3; a4; a6)
= {([x : y : z]; s) 2 P2x S | y2z + a1(s)xyz + a3(s)yz2 = x3+ a2(s)x2z + *
*a4(s)xz2+ a6(s)z3}
for some system of functions a1; : :;:a62 OS. (Whenever we write (a1; : :;:a6),*
* it is to be understood that
there is no a5.) For any such curve, there is an evident projection p:C -!S and*
* a section 0:S -!C given
by s 7! ([0 : 1 : 0]; s). We write WC(R) for the set of 5-tuples (a1; : :;:a6) *
*2 R5, which can clearly be
identified with the set of Weierstrass curves over spec(R). Thus, WC = spec(Z[a*
*1; : :;:a6]) is a scheme. We
define various auxiliary functions as follows:
b2= a21+ 4a2
b4= a1a3+ 2a4
b6= a23+ 4a6
b8= a21a6- a1a3a4+ 4a2a6+ a2a23- a24
c4= b22- 24b4
c6= -b32+ 36b2b4- 216b6
= -b22b8- 8b34- 27b26+ 9b2b4b6
j= c34=
The function 2 OS is called the discriminant. We say that a Weierstrass curve*
* C is smooth if its
discriminant is a unit in OS.
Definition B.2.A generalized elliptic curve over S is a scheme C equipped with *
*maps S 0-!C -p!S such
that S can be covered by open subschemes Sisuch that Ci= C xSSiis isomorphic to*
* a Weierstrass curve, by
an isomorphism preserving p and 0. An elliptic curve is a generalized elliptic *
*curve that is locally isomorphic
to a smooth Weierstrass curve. We shall think of S as being embedded in C as th*
*e zero-section. We write
!C=S for the cotangent space to C along S, or equivalently !C=S= IS=I2S, where *
*IS is the ideal sheaf of S.
One checks that this is a line bundle on S. We say that C=S is untwisted if !C=*
*S is trivializable.
It is possible to give an equivalent coordinate-free definition, but this req*
*uires rather a lot of algebro-
geometric machinery.
Let C be a Weierstrass curve. Note that if we put z = 0 then the defining equ*
*ation becomes x3 = 0, so
the locus where z = 0 is an infinitesimal thickening of the locus x = z = 0, wh*
*ich is our embedded copy of
S. Thus, the complementary open subscheme C1= C \ S is just the locus where z i*
*s invertible. This can be
identified with the curve in the affine plane with equation
y2+ a1xy + a3y = x3+ a2x2+ a4x + a6:
Weierstrass curves are often described by giving this sort of inhomogeneous equ*
*ation.
A given generalized elliptic curve can be isomorphic to two different Weierst*
*rass curves, and it is important
to understand the precise extent to which this can happen. For this, we define*
* a group scheme WR of
52 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
"Weierstrass reparameterizations": for any ring R, WR(R) is the group of matric*
*es of the form
0 1
u2 0 r
M(u; r; s) = @ su2 u3 tA
0 0 1
with u 2 Rx. Such a matrix acts by multiplication on P2x spec(R) in the obvious*
* way, and one checks that
it carries C(a1; : :;:a6) to C(a01; : :;:a06), where
a01= a1u - 2s
a02= a2u2+ a1su - 3r - s2
a03= a3u3- a1ru + 2rs - 2t
a04= a4u4+ a3su3- 2a2ru2+ a1(t - 2rs)u + 3r2+ 2rs2- 2st
a06= a6u6- a4ru4+ a3(t - rs)u3+ a2r2u2+ a1(r2s - rt)u + 2rst - t2- r2*
*s2- r3
(These equations are equivalent to [Del75, Equations 1.6] with aiand a0iexchang*
*ed.)
We therefore have an action of WR on WC, and a map from WRxWC to the scheme o*
*f triples (C; C0; f)
where C and C0are Weierstrass curves and f is an isomorphism C -!C0of pointed c*
*urves. One can check
that this map is an isomorphism.
If we define c04, c06, 0and j0in the obvious way then we have
c04= c4u4
c06= c6u6
0 = u12
j0= j:
Definition B.3.Let C be a generalized elliptic curve over S. We will define var*
*ious things as though C were
a Weierstrass curve; one can check that the definitions are local on S and inva*
*riant under reparameterization,
so they are well-defined in general. We write
Sell= D()
Ssing= V ()
Smult= D(c4) \ V ()
Sadd= V (c4) \ V ();
and call these the elliptic, singular, multiplicative and additive loci in S, r*
*espectively. Here as usual, D(a) is
the locus where a is invertible and V (a) is the locus where a = 0. Let f be a *
*standard Weierstrass equation
for C, and write fx = @f=@x and so on. Let Csingbe the closed subscheme of C wh*
*ere fx = fy = fz = 0,
and let Cregbe the complementary open subscheme.
It turns out that Creghas a unique structure as an abelian group scheme over *
*S such that the map
0: S -! Cregis the zero section. If C is a Weierstrass curve, then any three s*
*ections c0; c1; c2 of Creg
with c0+ c1+ c2 = 0 are collinear in P2, or equivalently the matrix formed by t*
*he coordinates of the ci
has determinant zero. Any map of generalized elliptic curves (compatible with *
*the projections and the
zero-sections) is automatically a homomorphism. One can check that the negation*
* map is given by
-[x : y : z] = [x : -a1x - y - a3z : z]:
The formal completion of C along S is written bC. If C is defined by a Weiers*
*trass equation f = 0 then
we have
bC(R) = {(x; z; s) 2 Nil(R)2x S(R) | f(x; 1; z) = 0};
where Nil(R) is the set of nilpotent elementsPin R. One checks using the formal*
* implicit function theorem that
there is a unique power series (x) = k>0kxk 2 OS[[x]] such that (x) = x3 (mod*
* x4), and (x; z; s) 2 bC(R)
if and only if z = (x). This proves that bC~=S x bA1, so that bCis a formal cur*
*ve over S. The rational
ELLIPTIC SPECTRA *
* 53
function x=y gives a coordinate; we normally work in the affine piece y = 1 so *
*this just becomes x. The
group structure on C thus makes bCinto a formal group over S (i.e. a commutativ*
*e, one-dimensional, smooth
formal group). If we define
X j
O(x0; x1; x2) = i+j+k+2xi0x1xk2
i;j;k0
then one can check that O(x0; x1; x2) = x0+ x1+ x2mod (x0; x1; x2)2 and
fifi fi
fifix01(x0) fifi
fifix11(x1) fifi= (x0- x1)(x0- x2)(x1- x2)O(x0; x1; x2):
x2 1 (x2) fi
One can deduce from this that O(x0; x1; x2) is a unit multiple of x0 +F x1 +F x*
*2, and that the series
G(x0; x1) = [-1]F(x0+F x1) is uniquely characterized by the equation O(x0; x1; *
*G(x0; x1)) = 0. We also
have
[-1]F(x) = -x=(1 + a1x + a3(x)):
More generally, if C is an untwisted generalized elliptic curve then bCis sti*
*ll a formal group, although we
do not have such explicit formulae in this case.
B.0.1. Modular forms.
Definition B.4.A modular form of weight k over Z is a rule g that assigns to ea*
*ch generalized elliptic curve
C=S a section g(C=S) of !kC=Sover S, in such a way that for each pull-back squa*
*re
"f
C ____//_C0
p || |p0|
fflffl|fflffl|
S __f_//_S0
of generalized elliptic curves, we have f*g(C0=S0) = g(C=S). (We will shortly c*
*ompare this with the classical,
transcendental definition.) We write MFk for the group of modular forms of weig*
*ht k over Z. More generally,
for any ring R, we define modular forms over R by the same procedure, except th*
*at S is required to be a
scheme over spec(R).
Let C = C(a1; : :;:a6) be the obvious universal Weierstrass curve over the sc*
*heme
WC = spec(Z[a1; : :;:a6]):
We have a projection map ss :WR x WC -!WC and also an action map ff:WR x WC -!W*
*C defined by
ff(a1; : :;:a6; r; s; t; u) = (a01; : :;:a06);
where the elements a0iare as in the previous section.
We can regard WR x C as a generalized elliptic curve over WR x WC, and we hav*
*e maps
"ss; "ff:WR x C -!C (B.*
*5)
covering ss and ff. The first of these is just the projection, and the second i*
*s given by the usual action of
WR < GL3 on P2. It is clear that the group of modular forms of weight k over Z *
*is precisely the set of
sections g(C=WC) of !kC=WCsuch that
ff*g(C=WC) = ss*g(C=WC): (B.*
*6)
More explicitly, there is the following.
Proposition B.7.The space MFk can be identified with the set of functions h 2 O*
*WC = Z[a1; : :;:a6] such
that ff*h = ukh. Moreover, we have an isomorphism of graded rings
MF* = Z[c4; c6; ]=(1728 - c34+ c26);
where c42 MF4, c62 MF6 and 2 MF12. (The prime factorization of 1728 is 2633.)
54 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
Proof.To understand the condition (B.6)more explicitly, we notice that x=y defi*
*nes a function on a neigh-
borhood of the zero-section in C, so we have a section d(x=y)0 of !C=WC , which*
* is easily seen to be a
basis. Moreover, we have ss*d(x=y)0 = d(x=y)0 and ff*d(x=y)0 = u-1d(x=y)0. Thus*
*, a section g(C=WC) of
!kC=WCis of the form g(C=WC) = h d(x=y)k0for a unique h 2 OWC = Z[a1; : :;:a6];*
* and equation (B.6)is
equivalent to the equation ff*h = ukss*h (and we implicitly identify ss*h with *
*h). It follows that c4, c6 and
correspond to modular forms of the indicated weights, and one checks directly *
*from the definitions that
c34- c26= 1728. The proof that MF* is precisely Z[c4; c6; ]=(1728 - c34+ c26) c*
*an be found_in [Del75] and
will not be reproduced here. *
* |__|
Definition B.8.The q-expansion of a modular form g is the series h(q) 2 Z[[q]] *
*= ODTatesuch that
g(CTate=DTate) = h(q)d(x=y)k0.
Note that if o lies in the upper half plane then the analytic variety Co = C=*
*Z{1; o} has a canonical
structure as a scheme over spec(C), which makes it an elliptic curve. Moreover,*
* if z is the obvious coordinate
on C, then the form dz on C gives an invariant differential on Co. Thus, for a*
*ny modular form g of
weight k we have a complex number f(o) such that g(Co= spec(C)) = f(o)(dz)k. If*
* abcd2 SL2(Z) and
o0= (ao +b)=(co +d) then multiplication by (co +d)-1 gives an isomorphism Co -!*
*Co0. The pull-back of dz
along this is (dz)=(co + d), so we conclude that f(o0) = (co + d)kf(o). One can*
* check that this construction
gives an isomorphism of C MF* with the more classical ring of holomorphic func*
*tions on the upper half
plane, satisfying the functional equation f(o0) = (co + d)kf(o) and a growth co*
*ndition at infinity. Moreover,
if g has q-expansion h(q) then the power series h(e2ssio) converges to f(o).
B.0.2. Invariant differentials. As Cregis a group scheme, the sections of !C=S *
*over S biject with the sec-
tions of 1C=Sover Cregthat are invariant under translation. This is proved by t*
*he same argument as the
corresponding fact for Lie groups. Another way to say this is as follows. A sec*
*tion of 1C=Sis the same as
a section of I =I2, where is the diagonal in CregxS Creg, and I is the associ*
*ated ideal sheaf. In other
words, it is a function ff(c0; c1) that is defined when c0 is infinitesimally c*
*lose to c1, such that ff(c; c) = 0.
In these terms, a section of the form g dh becomes the function (c0; c1) 7! g(c*
*0)(h(c0) - h(c1)). A section
of 1C=Sis invariant if and only if ff(c + c0; c + c1) = ff(c0; c1). On the othe*
*r hand, a section of !C=S is
a function fi(c) that is defined when c is infinitesimally close to 0, such tha*
*t fi(0) = 0. These biject with
invariant sections of 1C=Sby fi(c) = ff(c; 0) and ff(c0; c1) = fi(c0- c1).
We refer to invariant sections of 1C=Sas invariant differentials on C. We nex*
*t exhibit such a section when
C is a Weierstrass curve. Suppose that C is given by an equation f = 0, where
f(x; y; z) = y2z + a1xyz + a3yz2- x3- a2x2z - a4xz2- a6z3:
We write fx = @f=@x and so on. Next, observe that a point that is infinitesimal*
*ly close to 0 = [0 : 1 : 0]
has the form [ffl : 1 : 0] with ffl2 = 0. We need to calculate [x : 1 : z] + [*
*ffl : 1 : 0]. We know that
-[x : 1 : z] = [-x : 1 + a1x + a3z : -z] and -[ffl : 1 : 0] = [-ffl : 1 + a1ffl*
* : 0], and one checks that
fifi fi
fifi-ffl 1 + a1ffl 0 fifi 2
fifi-x 1 + a1x + a3z -z fifi= 0 (mod ffl )
x + fflfz 1 z - fflfxfi
and
f(x + fflfz; 1; z - fflfx) = 0 (mod ffl2):
This shows that
[x : 1 : z] + [ffl : 1 : 0] = [x + fflfz : 1 : z - fflfx] *
*(mod ffl2):
Thus, if we define a section fi0 of !C=S by fi0([ffl : 1 : 0]) = ffl, then the *
*corresponding invariant differential ff0
satisfies
ff0([x + fflfz : 1 : z - fflfx]; [x : 1 : z]) = ffl;
ELLIPTIC SPECTRA *
* 55
and thus ff0 = dx=fz. It is convenient to rewrite this in terms of homogeneous *
*coordinates: it becomes
ff0= y2d(x=y)=fz. We rewrite this again, and also introduce two further forms f*
*f1 and ff2, as follows:
ff0= y2d(x=y)=fz = (y dx - x dy)=fz
ff1= z2d(y=z)=fx = (z dy - y dz)=fx
ff2= x2d(z=x)=fy = (x dz - z dx)=fy:
We claim that any two of these forms agree wherever they are both defined. Inde*
*ed, one can check directly
that
ff0- ff1= (y df - 3f dy)=(fxfz)
ff1- ff2= (z df - 3f dz)=(fyfx)
ff2- ff0= (x df - 3f dx)=(fzfy);
and the right hand sides are zero because f = 0 on C and thus df = 0 on C. Thus*
*, we get a well-defined
differential form ff on the complement of the closed subscheme Csingwhere fx = *
*fy = fz = 0. We have seen
that ff0 is invariant wherever it is defined, and it follows by an evident dens*
*ity argument that ff is invariant
on all of Creg.
B.1. Examples of Weierstrass curves. In this section, we give a list of example*
*s of Weierstrass curves
with various universal properties or other special features. We devote the whol*
*e of the next section to the
Tate curve.
B.1.1. The standard form where six is invertible. Consider the curve C = C(0; 0*
*; 0; a4; a6) over the base
scheme S = spec(Z[1_6; a4; a6]) given by the equation
y2z = x3+ a4xz2+ a6z3;
equipped with the invariant differential
ff = -z_dx_+_x_dz2yz= y_dz_-_z_dy3x2+=a__2y_dx_-_x_dy_2: 2
4zy - 2a4xz - 3a6z
We have
c4= -243a4
c6= -2533a6
= -24(4a34+ 27a26)
j= 2833a34=(4a34+ 27a26)
This is the universal example of a generalized elliptic curve over a base where*
* six is invertible, equipped with
a generator ff of !E=S. More precisely, suppose we have a scheme S0where six is*
* invertible in OS0, and a
generalized elliptic curve C0-! S0. Suppose that the line bundle !C0=S0over S0i*
*s trivial, and that ff0is a
generator. Then there is a map f :S0-! S, and an isomorphism g :C0~=f*C, such t*
*hat the image of ff
under the evident map induced by f and g, is ff0. Moreover, the pair (f; g) is *
*unique.
Here is an equivalent statement: there is a unique quadruple (x0; y0; a04; a0*
*6) with the following properties:
i.x0and y0are functions on C01= C0\ S0.
ii.a4 and a6 are functions on S0.
iii.The functions x0and y0induce an isomorphism of C01with the curve (y0)2= (x*
*0)3+ a4x0+ a6 in A2x S.
iv.The form ff0|C01is equal to -dx0=(2y0).
56 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
B.1.2. The Jacobi quartic. The Jacobi quartic is given by the equation
Y 2= 1 - 2ffiX2 + fflX4
over Z[1_6; ffi; ffl]. The projective closure of this curve is singular, so ins*
*tead we consider the closure in P3 of its
image under the map [1; X; Y; X2]. This closure (which we will call C) is defin*
*ed by the equations
Y 2= W2 - 2ffiWZ + fflZ2
WZ = X2:
For generic ffi and ffl, the curve C is smooth and is the normalization of the *
*projective closure of the Jacobi
quartic. In all cases, C is isomorphic to the Weierstrass curve
y2z = (x - 12ffiz)((x + 6ffiz)2- 324ffl)
via
2)X2 + 2ffi(Y - 1))
x= 6((3ffl_-_ffi________Y + ffiX2 - 1
233(ffi2- ffl)X
y= 2__________Y + ffiX2 - 1
X = 6(12ffi - x)=y
Y= (2534ffi(ffi2- 3ffl) + 2333(ffi2+ 3ffl)x - 36ffix2+ y2)=*
*y2:
The standard invariant differential is as follows
ff = -dX=(6Y ) = -dx=(2y) = dy=(2233(ffi2+ 3ffl) - 3x2):
The zero section corresponds to the point
[W : X : Y : Z] = [1 : 0 : 1 : 0]:
There is also a distinguished point P of order two, given by
[W : X : Y : Z] = [1 : 0 : -1 : 0] or [x : y : z] = [12ffi*
* : 0 : 1]:
The curve C is the universal example of an elliptic curve with a given generato*
*r of !E and a given point of
order two, over a base scheme where six is invertible. Indeed, given such a cur*
*ve, the last example tells us
that there is a unique quadruple (x; y; a4; a6) giving an isomorphism of C with*
* the curve y2 = x3+ a4x + a6,
such that the given differential is d(x=y)0. The points of exact order two corr*
*espond to the points where
the tangent line is vertical. It follows that we must have y(P) = 0 and x(P) = *
*12ffi for some ffi, so that
123ffi3+12a4ffi +a6= 0, so x-12ffi divides x3+a4x+a6. As the coefficient of x i*
*n this polynomial is zero, one
checks that the remaining term has the form x2+12ffi+j for some j, or equivalen*
*tly the form (x+6ffi)2+324ffl
for some ffl. The claim follows easily from this.
The modular forms for the Jacobi curve are
c4= 2634(ffi2+ 3ffl)
c6= 2936ffi(ffi2- 9ffl)
= 212312(ffl - ffi2)2ffl
2+ 3ffl)3
j= 26(ffi____ffl(ffl:- ffi2)2
B.1.3. The Legendre curve. Consider the Weierstrass curve over Z[1_2; ] given by
y2z = x(x - z)(x - z):
ELLIPTIC SPECTRA *
* 57
The modular forms are
c4= 24(1 - + 2)
c6= 25( - 2)( + 1)(2 - 1)
= 242( - 1)2
j = 28(1 - + 2)3=(( - 1)22)
If we restrict to the open subscheme where and (1 - ) are invertible, then the*
* kernel of multiplication by
2 is a constant group scheme, with points
0 = [0 : 1 : 0] P = [0 : 0 : 1] Q = [1 : 0 : 1] P + Q = [*
* : 0 : 1]:
B.1.4. Singular fibers. The curve y2z + xyz = x3 is a nodal cubic, with multipl*
*icative formal group. There
is a birational map f from P1 to the curve, with inverse g:
f[s : t] = [st(s - t) : t2s : (s - t)3]
g[x : y : z] = [x + y : y]:
The map f sends 1 to [0 : 1 : 0], and sends both 0 and infinity to the singular*
* point [0 : 0 : 1]. If
s0s1s2 = 1 then the points f[s0 : 1], f[s1 : 1] and f[s2 : 1] are collinear, wh*
*ich shows that the restriction to
Gm = P1\ {0; 1} is a homomorphism. The discriminant is zero and the j invariant*
* is infinite.
The curve y2z = x3 is a cuspidal cubic, with additive formal group. There is *
*a birational map f from P1
to the curve, with inverse g:
f[s : t] = [t2s : t3: s3]
g[x : y : z] = [x : y]:
This sends infinity to the singular point [0 : 0 : 1] with multiplicity two, an*
*d sends 0 to [0 : 1 : 0]. If
s0+s1+s2= 0 then the points f[s0: 1], f[s1: 1] and f[s2: 1] are collinear, whic*
*h shows that the restriction
to Ga = P1\ {1} is a homomorphism. The discriminant is zero and the j invariant*
* is undefined.
B.1.5. Curves with prescribed j invariant. If a and b = a - 1728 are invertible*
* in R then we have a smooth
Weierstrass curve C over spec(R) with equation
y2z + xyz = x3- 36xz2=b - z3=b:
The associated modular forms are
c4= -c6= a=b
= a2=b3
j = a:
If 6 is invertible in R we can put a = 0 and get the singular curve (y + x=2)2 *
*= (x + 1=12)3, which has
c4= = 0 so that j is undefined.
B.2. Elliptic curves over C. Let C be an elliptic curve over C. It is well-know*
*n that there exists a complex
number o in the upper half plane and a complex-analytic group isomorphism C ~=C*
*o = C=, where is
the lattice generated by 1 and o. We collect here a number of formulae, which a*
*re mostly proved in [Sil94,
Chapter V] (for example). We write q = e2ssio, so the map z 7! u = e2ssizgives *
*an analytic isomorphism
Co ~=Cx=qZ. We also have an analytic isomorphism of Co with the curve
Y 2Z = 4X3 - g2XZ - g3Z3;
P
where gk = !2\0!-2k. The isomorphism is given by (z mod ) 7! ["(z) : "0(z) : *
*1], where
X
"(z) = z-2 + ((z - !)-2- !-2):
!2\0
58 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
This is to be interpreted as [0 : 1 : 0] if z lies in . We also have an analyti*
*c isomorphism of Co with the
Weierstrass curve
y2z + xyz = x3+ a4xz2+ a6z3;
where a4 and a6 are given by the same formulae as for the Tate curve in x2.6. T*
*his isomorphism sends
u = e2ssizto [x : y : 1], where x and y are again given by the same formulae as*
* for the Tate curve. We have
the following identities.
X = (2ssi)2(x + 1=12)
Y = (2ssi)3(2y + x)
a4 = -(2ssi)-4g2=4 + 1=48
a6 = -(2ssi)-6g3=4 - (2ssi)-4g2=48 + 1=1728:
B.3. Singularities.
Proposition B.9.Let C be a generalized elliptic curve over S. Then C is flat ov*
*er S.
Proof.We can work locally on S and thus assume that S is affine and that C is a*
* Weierstrass curve. Let C0
be the locus where z is invertible, which is isomorphic to the affine curve whe*
*re z = 1, which has equation
y2+ a1xy + a3y = x3+ a2x2+ a4x + a6. Thus, the ring of functions on C0 is a fre*
*e module of rank 2 over
OS[x], or of rank 3 over OS[y]. Either description makes it clear that OC0 is f*
*ree as a module over OS, so
C0 is flat over S. Similar arguments show that the locus C1 (where y is inverti*
*ble) is also flat. The union of
C0 and C1 is the complement of the closed subscheme where y = z = 0. On this lo*
*cus the defining equation
gives x3 = 0, which is impossible as x, y and z are assumed to generate the uni*
*t ideal. It_follows that
C0[ C1= C, and thus that C is flat over S. *
* |__|
Proposition B.10.The singular locus Csingis contained in the open subscheme C0=*
* C \S. The projection
p:C -!S sends Csinginto Ssing.
Proof.Our claims are local on S so we may assume that C is a Weierstrass cubic,*
* defined by an equation
f = 0 in the usual way. On S C we have x = z = 0 and y is invertible, so we ca*
*n take y = 1. We then
have fz = y2 = 1, so clearly S Cregand Csing C0.
Now consider a point P = [x : y : z] of C. If z = 0 then the defining equatio*
*n gives x3 = 0, so P lies in
an infinitesimal thickening of S C. It follows that C0 is the same as the comp*
*lementary open locus where
z is invertible.
Now consider a point P = [x : y : z] of Csing. By the above, z is invertible *
*so we may assume z = 1. We
can then shift our coordinates so that x = y = 0. This changes f but does not c*
*hange , as we see from the
standard transformation formulae. Let the new f be
f(x; y; 1) = (y2+ a1xy + a3y) - (x3+ a2x2+ a4x + a6):
We must have f(0; 0; 1) = fx(0; 0; 1) = fy(0; 0; 1) = 0, so a3 = a4 = a6 = 0. I*
*t follows that the parameters
bk are given by b2 = a21+ 4a2 and b4 = b6 = b8 = 0, and thus that = 0. In othe*
*r words,_P lies over Ssing
as claimed. *
* |__|
B.4. The cubical structure for the line bundle I(0) on a generalized elliptic c*
*urve. In this section
we give a proof of Proposition 2.55.
B.4.1. Divisors and line bundles. We will need to understand the relationship b*
*etween divisors and line
bundles in a form which is valid for non-Noetherian schemes. An account of div*
*isors on curves is given
in [KM85 ], but we need to genera-Lise this slightly to deal with divisors on C*
* xS C xS C over S, for
example. The issues involved are surely well-known to algebraic geometers, but *
*it seems worthwhile to have
a self-contained and elementary account.
Definition B.11.Let X be a scheme over a scheme S. An effective divisor on X o*
*ver S is a closed
subscheme Y X such that the ideal sheaf IY is invertible and the map Y -!S is *
*flat.
ELLIPTIC SPECTRA *
* 59
Suppose that S = spec(A) and X = spec(B) for some A-algebra B, and that Y = s*
*pec(B=b) for some
element b that is not a zero-divisor. Then IY corresponds to the principal idea*
*l Bb ~=B in B, and it is easy
to see that Y is a divisor if and only if B=b is a flat A-module.
Conversely, if Y is a divisor then one can cover S by open sets of the form S*
*0= spec(A) and the preimage
X0of S0by sets of the form spec(B) in such a way that Y \ spec(B) has the form *
*spec(B=b) as above.
Proposition B.12.Let Y and Z be effective divisors on X over S. Then there is a*
* unique effective divisor
Y + Z with IY +Z= IYIZ = IY OX IZ. The effective divisors form an abelian monoi*
*d Div+(X=S) under
this operation. Moreover, this monoid has cancellation.
Proof.We define Y + Z to be the closed subscheme defined by the ideal sheaf IYI*
*Z < OX . We claim that
the product map IY OX IZ -!IY +Z= IYIZ is an isomorphism. Indeed, the question *
*is local, and locally
it translates to the claim that Bb B Bc maps isomorphically to Bbc when b and c*
* are not zero-divisors, and
this claim is obvious. All that is left is to check that Y + Z is flat over S. *
*Locally, we have a short exact
sequence
B=b //c_//_B=bc_////_B=c
with B=b and B=c flat over A, so B=bc is also flat over A. The rest is clear. *
* |___|
Definition B.13.We write Div(X=S) for the group completion of the monoid Div+(X*
*=S), and refer to its
elements as divisors. The proposition implies that the natural map Div+(X=S) -!*
*Div(X=S) is injective.
It also implies that given a divisor Y = Y+ - Y-, we can define a line bundle I*
*Y = IY+I-1Y-and this is
well-defined up to canonical isomorphism.
Proposition B.14.Let f :X0 -!X be a flat map. Then the pull-back along f gives*
* a homomorphism
Div+(X=S) -!Div+(X0=S), with If*Y = f*IY as line bundles over X0. This extends *
*to give an induced
homomorphism f*: Div(X=S) -!Div(X0=S).
Proof.Let Y X be a divisor, and write Y 0= f*Y = Y xX X0. It is clear that thi*
*s is a closed subscheme
of X0. The induced map f0:Y 0-!Y is a pull-back of a flat map so it is again fl*
*at. The map Y -!S is flat
because Y is a divisor, so the composite Y 0-!S is flat. Let j :Y -! X and j0:Y*
* 0-!X0be the inclusion
maps. Essentially by definition we have f*OX = OX0 and f*j*OY = j0*(f0)*OY = j0*
**OY 0. We have a short
exact sequence of sheaves IY -!OX -!j*OY, where j :Y -!X is the inclusion. As f*
* is flat, the functor f*
is exact, so we have a short exact sequence f*IY -!OX0 -!j0*OY 0. It follows th*
*at IY 0= f*IY, and f*IY
is clearly a line bundle. Thus, Y 0is a divisor, as required. It is easy to see*
* that f* is a homomorphism,_and
it follows by general nonsense that it induces a map of group completions. *
* |__|
Proposition B.15.Let g :S0-! S be an arbitrary map, and write X0= g*X. Then pul*
*l-back along g gives
a homomorphism Div+(X=S) -!Div+(X0=S0), with Ig*Y= g*IY as line bundles over X0*
*. This extends to
give an induced homomorphism f*: Div(X=S) -!Div(X0=S).
Proof.The proof is similar to that of the previous result. *
* |___|
Definition B.16.Let L be a line bundle over X, and u a section of L. Then ther*
*e is a largest closed
subscheme Y of X such that u|Y = 0. If this is a divisor, we say that u is divi*
*sorial and write div(u) = Y .
If so, then u is a trivialization of the line bundle L IY, so L ~=I-1Y.
If v is a divisorial section of another line bundle M then one can check that*
* u v is a divisorial section
of L M with div(u v) = div(u) + div(v). One can also check that the formation*
* of div(u) is compatible
with the two kinds of base change discussed in Propositions B.14 and B.15.
Definition B.17.A meromorphic divisorial section u of a line bundle L is an exp*
*ression of the form
u+=u-, where u+ and u- are divisorial sections of line bundles L+ and L- with a*
* given isomorphism
L = L+=L-. These expressions are subject to the obvious sort of equivalence rel*
*ation. We define div(u) =
div(u+) - div(u-), which is well-defined by the above remarks. We again have L *
*~=I-1div(u).
60 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
Lemma B.18. Let C be a subscheme of P2x S defined by a single homogeneous equat*
*ion f = 0 of degree
m, such that the coefficients of f generate the unit ideal in OS. Let Cregbe th*
*e open subscheme D(fx) [
D(fy) [ D(fz) of C, where fx, fy and fz are the partial derivatives of f. Let o*
*e be a section of Cregover S.
Then oeS C is a divisor.
Proof.Let U, V and W be the open subschemes of S where fxO oe, fyO oe and fzO o*
*e are invertible. Because
oe is a section of Cregwe know that S = U [ V [ W. We restrict attention to U; *
*a similar argument can
be given for V and W. After replacing S by U, we may assume that fx O oe is inv*
*ertible. Let C1 and C2
be the open subschemes where y and z are invertible. Because f is homogeneous *
*of degree m we have
xfx + yfy + zfz = mf and f O oe = 0 so x = -yfy=fx - zfz=fx on the image of oe.*
* Thus, on the closed
subscheme where y = z = 0 we also have x = 0, so this subscheme is empty, which*
* implies that C = C1[C2.
Write Ui= oe-1Ci, so that U = U1[ U2. We restrict attention to U2; a similar ar*
*gument can be given for
U1. In this context we can work with the affine plane where z = 1, and x, y and*
* f can be considered as
genuine functions. Write x0 = x O oe and y0 = y O oe. As f O oe = 0 we have f =*
* (x - x0)g + (y - y0)h for
some functions g and h. Clearly, g(x0; y0) = fx(x0; y0) and this is assumed inv*
*ertible, so D(g) is an open
subscheme of C2 containing oeU2. On this scheme we have f = 0 and thus x = x0- *
*(y - y0)h=g. Thus
D(g) \ V (y - y0) = D(g) \ V (x - x0; y - y0) = D(g) \ oeS:
Thus, in the open set D(g), our subscheme oeS is defined by a single equation y*
* = y0, so the corresponding
ideal sheaf is generated by y - y0.
We still need to verify that y - y0 is not a zero-divisor on D(g) \ C2. It is*
* harmless to shift coordinates so
that y0= x0= 0. Suppose that r 2 OS[x; y] is such that ry = 0 on D(g)\C2; we ne*
*ed to show that r = 0 on
D(g)\C2. We have gkry = sf in OS[x; y] for some k and s. It follows that gk+1rx*
* = gkr(f-hy) = (gkr-hs)f
and thus (gkr - hs)yf = gk+1rxy = gsxf. As the coefficients of f generate OS we*
* know that f is not a
zero-divisor in OS[x; y] so (gkr - hs)y = gsx. It follows easily that y divides*
* gs, say gs = ty, and then
gk+1ry = gsf = tfy so gk+1r = tf. On C2 we have f = 0 and thus gk+1r = 0, so on*
* D(g) \ C2 we have
r = 0 as required.
This shows that the intersection of oeS with D(g) \ C2 is a divisor. Similar *
*arguments cover the rest of
oeS with open subschemes of C in which oeS is a divisor. Trivially, the (empty)*
* intersection of oeS with_the
open subscheme C \ oeS is a divisor. This covers the whole of C, as required. *
* |__|
Corollary B.19.If C is a generalized elliptic curve over S then the zero sectio*
*n of C is a divisor. |___|
B.4.2. The line bundle I(0). Let C be a generalized elliptic curve over S, and *
*let I(0) denote the ideal sheaf
of S C. The smooth locus Cregis a group scheme over S, so we can define 3(I(0)*
*) over Cregand thus
the notion of a cubical structure. In this section we give a divisorial formula*
* for 3(I(0)).
Consider the scheme C3S= C xS C xS C. A typical point of C3Swill be written a*
*s (c0; c1; c2). We write
[c0 = c1] for the largest closed subscheme of (Creg)3Son which c0 = c1, and so *
*on. This is the pull-back of
the divisor S Cregunder the map g :(c0; c1; c2) 7! c0- c1. This map is the com*
*posite of the isomorphism
(c0; c1; c2) -!(c0- c1; c1; c2) with the projection map (Creg)3S-!Creg, and the*
* projection is flat because C
is flat over S (Proposition B.9). Thus, g is flat. It follows from Proposition *
*B.14 that [c0 = c1] is a divisor,
and the associated ideal sheaf is g*I(0). Similar arguments show that the subsc*
*hemes [ci= 0], [ci= cj],
[ci+ cj= 0] and [c0+ c1+ c2= 0] are all divisors (assuming that i 6= j). We can*
* thus define divisors
D1 = [c0= 0] + [c1= 0] + [c2= 0]
D2 = [c0+ c1= 0] + [c1+ c2= 0] + [c2+ c0= 0]
D3 = [c0+ c1+ c2= 0]
D4 = [c0= c1] + [c1= c2] + [c2= c0]:
There is (almost by definition) a canonical isomorphism of line bundles
3(I(0)) = I(0)0I-D1+D2-D3= !CID2I-1D1+D3:
ELLIPTIC SPECTRA *
* 61
B.4.3. A formula for the cubical structure.
Definition B.20.Let C = C(a1; a2; a3; a4; a6) be a Weierstrass curve. A typical*
* point of (Creg)3Swill be
written as (c0; c1; c2), with ci= [xi: yi: zi]. We define s(a_) by the followin*
*g expression:
fifi fi-1
fix0 y0 z0fifififix0z0fifififix1z1fifififix2z2fifi
s(a_)(c0; c1; c2) = fifix1y1z1fifififi fifififififififififi(z0z1z2)-1d*
*(x=y)0:
fix2 y2 z2fi x1 z1 x2 z2 x0 z0
Proposition B.21 (2.55).s(a_) is a meromorphic divisorial section of the line b*
*undle p*!C over (Creg)3S
(where p:C3S-!S is the projection). Its divisor is -D1+ D2- D3 (in the notation*
* of xB.4.2), so it defines
a trivialization of
(p*!C) I-D1+D2-D3= 3(I(0));
which is equal to s(C=S).
Proof.By an evident base-change, we may assume that C is the universal Weierstr*
*ass curve over S =
spec(Z[a1; a2; a3; a4; a6]), and thus that S is a Noetherian, integral scheme.
We have a bundle O(1) over C, whose global sections are homogeneous linear fo*
*rms in x, y and z. We
can take the external tensor product of three copies of O(1) to get a bundle L *
*over C xS C xS C. We define
a section u of L by
fifi fi
fix0 y0 z0fifi
u(c0; c1; c2) = fifix1y1z1fifi:
fix2 y2 z2fi
We claim that this is divisorial, and that div(u) = D4+ D3. This is plausible, *
*because one can easily check
that u = 0 on the divisors [ci= cj] (whose sum is D3) and also on the divisor [*
*c0+ c1+ c2= 0] (because any
three points that sum to zero are collinear). Let U0 be the open subscheme of (*
*Creg)3Swhere c1 6= c2, and
define U1 and U2 similarly. Then the complement of U = U0[U1[U2 is the locus wh*
*ere c0= c1= c2, which
has codimension 2. Given this, it is enough to check that u|Uiis divisorial and*
* that div(u|Ui) = (D4+D3)\Ui
for 0 i 2 (see [Har77, Proposition II.6.5]). By symmetry, we need only consid*
*er the case i = 0. Let
V0 be the complement of the diagonal in (Creg)2S, so that U0 = CregxS V0, which*
* we can think of as the
regular part of a generalized elliptic curve over V0. The diagonal is defined b*
*y the vanishing of the quantities
x1y2- x2y1, y1z2- y2z1, and z1x2- z2x1, so on V0 these quantities generate the *
*unit ideal. It follows from
this that the map
h:[s1: s2] 7! [s1x1+ s2x2: s1y1+ s2y2: s1z1+ s2z2]
gives an isomorphism of P1 with the locus in P2 where the determinant vanishes.*
* The addition law on C is
defined by the requirement that the intersection of h(P1) with CxSV0is [c0= c1]*
*+[c0= c2]+[c0= -c1-c2].
Moreover, we have [c1= c2] \ U0= ;. Thus, div(u) \ U0= (D4+ D3) \ U0 as require*
*d.
We now define sections v and w of L and L2 by
v(c0; c1;=c2)z0z1z2
fifi fifififififififififi
w(c0; c2;=c2)fifix0z0xfifififix1z1fifififix2z2fifi:
z1 x21 z2 x0 z0
By methods similar to the above, we find that
div(z0) = 3[c0= 0]
div(x0z1- x1z0) = [c0= 0] + [c1= 0] + [c0= c1] + [c0+ c1= 0]
and thus
div(v)= 3D1
div(w)= 2D1+ D2+ D4:
We also have s(a_) = u-1wv-1d(x=y)0 so as claimed this is a meromorphic divisor*
*ial section of p*!C, with
divisor -D1+ D2- D3. As explained earlier, it therefore gives rise to a trivial*
*ization of 3(I(0)).
62 M. ANDO, M. J. HOPKINS, AND N. P. STRICKLAND
Recall that 3(I(0)) is canonically trivialized on the locus where c2 = 0. In*
* terms of our picture of
3(I(0)) involving rational one-forms, this isomorphism sends a one-form to its *
*residue at c2 = 0. To
calculate this for s(a_), we may as well restrict attention to the affine piece*
* where y0 = y1 = y2 = 1, and let
x2 tend to zero. The 3 x 3 determinant in the definition of s(a_) approaches - *
*|x0z0x1z1|. The defining cubic
gives the relation
z2(1 + a1x2- a2x22+ a3z2- a4x2z2- a6z22) = x32;
which shows that z2is asymptotic to x32and thus that |x1z1x2z2|is asymptotic to*
* -x2z1and |x2z2x0z0|is asymptotic
to x2z0. (Here we say that two functions f and g are asymptotic if there is a f*
*unction h on a neighborhood
of the locus c2 = 0 such that f = gh and h = 1 when c2 = 0). It follows that s(*
*a_)(c0; c1; c2) is asymptotic
to x-12d(x)0, and this means that s(a_) has residue 1, as required.
We now see that s(a_) is a rigid section of 3(I(0)), so that f = s(a_)=s(C=S)*
* is an invertible function on
(Creg)3S, whose restriction to S is 1. It follows that f = 1 on the open subsch*
*eme p-1Sell, which_is dense in
(Creg)3S, so f = 1 everywhere. Thus s(a_) = s(C=S). *
* |__|
References
[ABS64] Michael F. Atiyah, Raoul Bott, and Arnold Shapiro. Clifford modules. To*
*pology, 3 suppl. 1:3-38, 1964.
[Ada76] J. Frank Adams. Primitive elements in the K-theory of BSU. Quart. J. Ma*
*th. Oxford Ser. (2), 27(106):253-262,
1976.
[Ada74] J. Frank Adams. Stable Homotopy and Generalised Homology. University of*
* Chicago Press, Chicago, 1974.
[AS98] Matthew Ando and Neil P. Strickland. Weil pairings and Morava K-theory.*
* 23 pp., To appear in Topology, 1998.
[Bre83] L. Breen. Fonctions Theta et Theoreme du Cube, volume 980 of Lecture No*
*tes in Mathematics. Springer-Verlag,
1983.
[BT89] Raoul Bott and Clifford Taubes. On the rigidity theorems of Witten. J. *
*Amer. Math. Soc., 2(1):137-186, 1989.
[Del75] Pierre Deligne. Courbes elliptiques: formulaire (d'apres J. Tate). In M*
*odular Functions of One Variable III, volume
476 of Lecture Notes in Mathematics, pages 53-73. Springer-Verlag, 1975.
[Dem72] Michel Demazure. Lectures on p-divisible groups. Springer-Verlag, Berli*
*n, 1972. Lecture Notes in Mathematics, Vol.
302.
[DG70] Michel Demazure and Pierre Gabriel. Groupes algebriques. Tome I: Geomet*
*rie algebrique, generalites, groupes
commutatifs. Masson & Cie, Editeur, Paris, 1970. Avec un appendice Corp*
*s de classes local par Michiel Hazewinkel.
[DR73] P. Deligne and M. Rapoport. Les schemas de modules de courbes elliptiqu*
*es. In Modular Functions of One Variable
II, volume 349 of Lecture Notes in Mathematics, pages 143-316. Springer*
*-Verlag, 1973.
[EKMM96]A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May. Rings, Modules *
*and Algebras in Stable Homotopy Theory,
volume 47 of Amer. Math. Soc. Surveys and Monographs. American Mathemat*
*ical Society, 1996.
[Fra92] J. Franke. On the construction of elliptic cohomology. Math. Nachr., 15*
*8:43-65, 1992.
[FS80] G. Frobenius and L. Stickelberger. Uber die Addition und Multiplication*
* der elliptischen Functionen. J. f"ur die
reine u. angewandte Math., 88:146-184, 1880. Reproduced in Frobenius, O*
*euvres Completes, Springer 1968.
[Gro72] Groupes de monodromie en geometrie algebrique. I. Springer-Verlag, Berl*
*in, 1972. Seminaire de Geometrie
Algebrique du Bois-Marie 1967-1969 (SGA 7 I), Dirige par A. Grothendiec*
*k. Avec la collaboration de M. Ray-
naud et D. S. Rim, Lecture Notes in Mathematics, Vol. 288.
[Har77] Robin H. Hartshorne. Algebraic Geometry, volume 52 of Graduate Texts in*
* Mathematics. Springer-Verlag, 1977.
[Haz78] Michiel Hazewinkel. Formal Groups and Applications. Academic Press, 197*
*8.
[HBJ92] Friedrich Hirzebruch, Thomas Berger, and Rainer Jung. Manifolds and mod*
*ular forms. Aspects of Mathematics,
E20. Friedr. Vieweg & Sohn, Braunschweig, 1992. With appendices by Nils*
*-Peter Skoruppa and by Paul Baum.
[HM98] M. J. Hopkins and M. Mahowald. Elliptic curves and stable homotopy theo*
*ry II. in preparation, 1998.
[HMM98] M. J. Hopkins, M. Mahowald, and H. R. Miller. Elliptic curves and stabl*
*e homotopy theory I. in preparation, 1998.
[Hop95] Michael J. Hopkins. Topological modular forms, the Witten genus, and th*
*e theorem of the cube. In Proceedings of
the International Congress of Mathematicians, Vol. 1, 2 (Z"urich, 1994)*
*, pages 554-565, Basel, 1995. Birkh"auser.
[Hus75] Dale Husemoller. Fibre Bundles, volume 33 of Graduate Texts in Mathemat*
*ics. Springer-Verlag, 1975.
[Jac] C. G. J. Jacobi. Formulae novae in theoria transcendentium ellipticarum*
* fundamentales. Crelle J. f"ur die reine u.
angewandte Math., 15:199-204. Reproduced in Gesammelte Werke Vol. I 335*
*-341, Verlag von G. Reimer, Berlin
1881.
[Kat73] Nicholas M. Katz. p-adic properties of modular schemes and modular form*
*s. In Modular functions of one variable,
III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), page*
*s 69-190. Lecture Notes in Mathematics,
Vol. 350, Berlin, 1973. Springer.
[KM85] N. M. Katz and B. Mazur. Arithmetic Moduli of Elliptic Curves, volume 1*
*08 of Annals of Mathematics Studies.
Princeton University Press, 1985.
[Lan87] Serge Lang. Elliptic functions, volume 112 of Graduate Texts in Mathema*
*tics. Springer-Verlag, New York, second
edition, 1987. With an appendix by J. Tate.
[Liu95] Kefeng Liu. On modular invariance and rigidity theorems. J. Differentia*
*l Geom., 41(2):343-396, 1995.
ELLIPTIC SPECTRA *
* 63
[LRS95] Peter S. Landweber, Douglas C. Ravenel, and Robert E. Stong. Periodic c*
*ohomology theories defined by elliptic
curves. In The Cech centennial (Boston, MA, 1993), volume 181 of Contem*
*p. Math., pages 317-337. Amer. Math.
Soc., Providence, RI, 1995.
[MR81] Mark Mahowald and Nigel Ray. A note on the Thom isomorphism. Proc. Amer*
*. Math. Soc., 82(2):307-308, 1981.
[MM65] John W. Milnor and John C. Moore. On the structure of Hopf algebras. An*
*nals of Mathematics, 81(2):211-264,
1965.
[Mor89] Jack Morava. Forms of K-theory. Math. Z., 201(3):401-428, 1989.
[Mum65] David Mumford. Biextensions of formal groups. In Arithmetic algebraic g*
*eometry (proceedings of Purdue confer-
ence). Harper, 1965.
[Mum70] David Mumford. Abelian Varieties, volume 5 of Tata institute of fundame*
*ntal research series in mathematics.
Oxford University Press, 1970.
[Och87] S. Ochanine. Sur les genres multiplicatifs definis par des integrals el*
*liptiques. Topology, 26:143-151, 1987.
[Qui69] Daniel G. Quillen. On the formal group laws of unoriented and complex c*
*obordism. Bulletin of the American
Mathematical Society, 75:1293-1298, 1969.
[Qui71] Daniel G. Quillen. The spectrum of an equivariant cohomology ring, I an*
*d II. Annals of Mathematics, 94:549-602,
1971.
[Ros98] Ioanid Rosu. Equivariant elliptic cohomology and rigidity. PhD thesis, *
*MIT, 1998.
[RW77] Douglas C. Ravenel and W. Stephen Wilson. The Hopf ring for complex cob*
*ordism. Journal of Pure and Applied
Algebra, 9:241-280, 1977.
[Seg88] Graeme Segal. Elliptic cohomology. In Seminaire Bourbaki 1987/88, volum*
*e 161-162 of Asterisque, pages 187-201.
Societe Mathematique de France, Fevrier 1988.
[Sil86] Joseph H. Silverman. The Arithmetic of Elliptic Curves. Graduate Texts *
*in Mathematics. Springer-Verlag, New
York, 1986.
[Sil94] Joseph H. Silverman. Advanced Topics in the Arithmetic of Elliptic Curv*
*es, volume 151 of Graduate Texts in
Mathematics. Springer-Verlag, New York, 1994.
[Sin68] William M. Singer. Connective fiberings over BU and U. Topology, 7, 196*
*8.
[Str99a]Neil P. Strickland. Formal schemes and formal groups. In J.P. Meyer, J.*
* Morava, and W.S. Wilson, editors,
Homotopy-invariant algebraic structures: in honor of J.M. Boardman, Con*
*temporary Mathematics. American
Mathematical Society, 1999. 75 pp., To Appear.
[Str99b]Neil P. Strickland. Products on MU-modules. Transactions of the America*
*n Mathematical Society, 351:2569-2606,
1999.
[Tat74] John T. Tate. The arithmetic of elliptic curves. Invent. Math., 23:179-*
*206, 1974.
[Wit87] Edward Witten. Elliptic genera and quantum field theory. Comm. Math. Ph*
*ysics, 109:525-536, 1987.
[Wit88] Edward Witten. The index of the Dirac operator in loop space. In P. S. *
*Landweber, editor, Elliptic Curves and
Modular Forms in Algebraic Topology, volume 1326 of Lecture Notes in Ma*
*thematics, pages 161-181, New York,
1988. Springer-Verlag.
Department of Mathematics, The University of Illinois at Urbana-Champaign, Ur*
*bana IL 61801, USA
E-mail address: mando@math.uiuc.edu
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, *
*MA 02139-4307, USA
E-mail address: mjh@math.mit.edu
Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH, En*
*gland
E-mail address: N.P.Strickland@sheffield.ac.uk
*