Beta-elements and divided congruences
Jens Hornbostel and Niko Naumann
Abstract
The f-invariant is an injective homomorphism from the 2-line of the Adams-N*
*ovikov spec-
tral sequence to a group which is closely related to divided congruences of*
* elliptic modular
forms. We compute the f-invariant for two infinite families of fi-elements *
*and explain the
relation of the arithmetic of divided congruences with the Kervaire invaria*
*nt one problem.
1. Introduction
One of the most successful tools for studying the stable homotopy groups of sph*
*eres is the Adams-
Novikov spectral sequence (ANSS)
Es,t2= Exts,tMU*MU(MU *, MU *) ) ssst-s(S0).
The corresponding filtration on sss*:= sss*(S0) defines a succession of invaria*
*nts of framed bordism,
each being defined whenever all of its predecessors vanish, the first one of wh*
*ich is simply the degree
d : F0,*=F1,*= sss0-! E0,02= Z
which is an isomorphism. The next invariant, defined for all n > 0, is the e-in*
*variant
e : sssn= F1,n+1-! E1,n+12 Q=Z,
c.f. [Sw , Chapter 19]. Though defined purely homotopy-theoretic here, the e-in*
*variant is well-known
to encode subtle geometric information. For its relation to index theory via th*
*e j-invariant see [APS ,
Theorem 4.14]. The e-invariant vanishes for all even n = 2k > 2, thus giving ri*
*se to the f-invariant
f : sss2k= F2,2k+2-! E2,2k+22.
The understanding of this invariant is fragmentary at the moment. In particular*
*, there is no index-
theoretic interpretation of it comparable to the one available for the e-invari*
*ant.
As a first step towards understanding the f-invariant, G. Laures [L1] showed ho*
*w elliptic homology
can be used to consider the f-invariant
_______________________________________________________________________________*
*__
2000 Mathematics Subject Classification 55Q10, 55T15, 55N34, 11F33
Keywords: beta elements, f-invariant, divided congruences
Jens Hornbostel and Niko Naumann
f : sss2k-! E2,2k+22,! D__k+1 Q=Z
as taking values in a group which is closely related to divided congruences of *
*modular forms. Note
that this is similar to the role taken by complex K-theory in the study of the *
*e-invariant. Strictly
speaking, at this point we had better switched from MU to BP. In fact, we will *
*always work locally
at a fixed prime p in the following.
This surprising connection of stable homotopy theory with something as genuinel*
*y arithmetic as
divided congruences certainly motivates to ask for a thorough understanding of *
*how these are related
by the f-invariant.
The main purpose of this paper is to make this relation explicit.
We also include a fairly self-contained review of G. Laures' above version of t*
*he f-invariant to help
the reader who might be interested in making his own computations. We now revie*
*w the individual
sections in more detail.
In section 2, we first remind the reader of the fi-elements which generate the *
*2-line of the ANSS
(with a little exception at the prime 2). We then construct, for suitable Hopf *
*algebroids, a complex
which is quasi-isomorphic to the cobar complex and which will facilitate later *
*computations. Finally,
we show how to use elliptic homology to obtain the f-invariant as above.
In section 3, we recall some fundamental results of N. Katz on the arithmetic o*
*f divided congruences
and point out an interesting relation between BP-theory and the mod p Igusa tow*
*er (Theorem 5).
Next, we give some specific computations of modular forms and divided congruenc*
*es for 1(3) which
n*
*+2
we will need to study the f-invariant of the Kervaire elements fi2n,2n2 Ext2,2 *
*[BP ] at the prime
p = 2.
In section 4, we first compute the f-invariants of the infinite family of fi-el*
*ements fit for t > 1 not
divisible by p (Theorem 10). Then we explain how to approach the problem of com*
*puting the Chern
numbers determining the fi2n,2n(see [L2, Corollary 4.2.5] for the case of fi1 a*
*t the prime 3). We do
this by explicit computations in BP-theory for dimension 2 and 6 (Theorem 18). *
*The computations
get very complicated in higher dimensions. In order to use divided congruences,*
* we then compute
the f-invariants of the family fis2n,2nfor n > 0 and s > 1 odd (Theorem 19). We*
* hope that clever
use of divided congruences will enable us to compute the Chern numbers determin*
*ing the fi2n,2nfor
all n. See Corollary 21 for a quick summary of what we can and cannot do at the*
* moment.
Acknowledgements
The idea that it might be possible to project to the element fi2n,2nusing divid*
*ed congruences - and
consequently rephrase the Kervaire invariant one problem - was communicated to *
*us by G. Laures.
2.The construction of the f-invariant
We remind the reader of the construction of fi-elements in section 2.1. In sect*
*ion 2.2, we construct
a complex which is quasi-isomorphic to the cobar complex and in which we will c*
*ompute represen-
tatives for some of the fi-elements. This is used in section 2.3 where we expla*
*in how to express the
f-invariant of elements in Ext2,2k[BP ] in terms of divided congruences.
2
Beta-elements and divided congruences
2.1 Beta-elements in stable homotopy
We review some facts on Brown-Peterson homology BP at the prime p and fi-elemen*
*ts. See [MRW ]
and [R ] for more details. The Brown-Peterson spectrum BP has coefficient ring *
*BP* = Z(p)[v1, v2, ...]
with vi in dimension 2(pi- 1). The universal p-typical formal group law is defi*
*ned over this ring.
The couple (A, ) := (BP *, BP*BP ) becomes a Hopf algebroid in a standard way *
*and we have
BP *BP = BP* Z(p)[t1, t2, ...] such that the left unit jL of the Hopf algebroid*
* (A, ) is the standard
inclusion. The right unit jR is determined over Q by the formula in [R , Theore*
*m A.2.1.27]. Choosing
the Hazewinkel generators [R , A.2.2.1] for the vi, a short computation yields *
*jR(v1) = v1+ pt1 and
jR(v2) = v2+ v1tp1- vp1t1 mod p.
We have the chromatic resolution of BP* as a left BP*BP -comodule
BP*! M0 ! M1 ! ...
which gives rise to the chromatic spectral sequence
Exts,*BP*BP(BP *, Mt) ) Exts+t,*BP*BP(BP *, BP*).
This allows the construction of elements in Ext*,*BP*BP(BP *, BP*), the so call*
*ed Greek-letter elements.
Strictly speaking, these elements arise from comodule sequences 0 ! Nn ! Mn ! N*
*n+1 ! 0,
but for our computations we will need the related comodule sequences (1) and (2*
*) below. See
[MRW , Lemma 3.7 and Remark 3.8] for the relationship between them. We abbrevi*
*ate Hn(.) :=
Extn,*BP*BP(BP *, .) in the following.
To construct the fi-elements [MRW , p. 476/477], choose integers t, s, r > 1 s*
*uch that (pr, vs1, xt0n)
BP *is an invariant ideal where t = pnt0, (p, t0) = 1 and xn is a homogeneous p*
*olynomial in v1, v2
and v3 considered as an element of v-12BP*=(pr, vs1) (see [R , p. 202] or [MRW *
* , p. 476]), for example
x0 = v2.
Consider the two short exact sequences of BP*BP -comodules
r
(1) 0 ! BP* p!BP *! BP*=(pr) ! 0
vs1 r r s
(2) 0 ! 2s(p-1)BP*=(pr) ! BP *=(p ) ! BP*=(p , v1) ! 0.
Using the induced boundary maps
ffi : H0(BP *=(pr, vs1)) -! H1(BP *=(pr)) and
ffi0: H1(BP *=(pr)) -! H2(BP *)
we define fit,s,r:= ffi0ffi(xt0n).
It is known [MRW , Theorem 2.6], [Sh] for which indices (t, s, r) the elements*
* fit,s,rare non-zero in
H2(BP*). In this case the order of fit,s,ris pr. By construction, we have fit,s*
*,r2 H2,2t(p2-1)-2s(p-1)(BP *).
n+2
Example 1. For p = 2 and n > 1, the element fi2n,2n:= fi2n,2n,12 Ext2,2BP*BP(BP*
* *, BP*) is called the
Kervaire element. It is mapped via the Thom reduction [R , Theorem 5.4.6] to th*
*e element h2n+12
n+2
Ext2,2HZ=2*HZ=2(HZ=2*, HZ=2*). The latter element survives to a non-zero elemen*
*t of ss2n+2-2(S0) in
the Adams spectral sequence if and only if (as W. Browder [B , Theorem 7.1] has*
* shown) there exists
3
Jens Hornbostel and Niko Naumann
a framed manifold of dimension 2n+2 - 2 with non-vanishing Kervaire invariant. *
*Whether or not
this is the case is unknown for n > 5, for n 6 4 see [BJM ], [KM ].
2.2 The rationalised cobar complex
The standard way of displaying elements in Extn := Extn(A, A) of a flat Hopf al*
*gebroid (A, ) is
by means of the cobar complex. In this section we shall give another descriptio*
*n of this Ext group
as a subquotient of (A Q) n needed to compute f-invariants (Proposition 3). T*
*he results of this
section are a more algebraic version of [L1, Section 3.1].
Let (A, ) be a Hopf algebroid with structure maps jL, jR, ffl and . This dete*
*rmines a cosimplicial
abelian group .as follows: Set n := An with cofaces @i : n -! n+1; @0(fl1*
* . . .fln) :=
1 fl1 . . .fln; @i(fl1 . . .fln) := fl1 . . . (fli) . . .fln (1 6 i 6 n) and @n*
*+1(fl1 . . .fln) :=
fl1 . . .fln 1 for n > 1 and @0 := jR, @1 := jL for n = 0 and codegeneracies*
* oei: n+1 -! n,
oei(fl0 . . .fln) := fl0 . . .ffl(fli) . . .fln. We also denote by .the as*
*sociated cochain complex.
Following [R , Definition A.1.2.11], we define the reduced cobar complex (usual*
*ly denoted as C.(A, A))
__. __n __ An __ *
* __0
as being the subcomplex .with := for n_> 1_where := ker(ffl)_and*
* := A. This
is a subcomplex because (fl) - fl 1 - 1 fl 2 A for any fl 2 .
We now assume that (A, ) is a flat Hopf algebroid such that
i) A (and hence ) is torsion free.
ii) The map AQ2 := (A Q) 2 -OE! Q := Q, a b 7! a.jR(b) is an isomorphism (jL *
*is suppressed
from notation).
iii) The Q-algebra AQ is augmented by some o : AQ -! Q.
Remark 2. The above assumptions i)-iii) are fulfilled by the flat Hopf algebroi*
*d (BP *, BP*BP ):
The proof of ii) follows from the fact that over any Q-algebra any two (p-typic*
*al) formal group
laws are isomorphic via a unique strict isomorphism. If BP* -! E* is a non-zero*
* Landweber exact
algebra [HS ], then (E*, E*E) also fulfils all the above assumptions: E* is a Z*
*(p)-algebra on which p
is not a zero divisor, hence E* is torsion free. Conditions ii), iii) and the f*
*latness of of (E*, E*E) are
inherited from (BP *, BP*BP ) because E*E ' E* BP* BP*BP BP* E* , c.f. [Na , *
*Proposition 10]
for the flatness.
We define the cosimplicial abelian group D.by Dn := AQ(n+1)(n > 0) with cofaces*
* @i : Dn -!
Dn+1 to be given by @i(a0 . . .an) := a0 . . .1 . . .an with the 1 in the*
* (i + 1)st
position (i = 0, . .,.n + 1) and codegeneracies oei : Dn -! Dn-1 (i = 0, . .,.,*
* n - 1) defined by
oei(a0 . . .an) := a0 . . .aiai+1 . . .an. By ii) above we have for any n > *
*0 an isomorphism
n A n
(3) OEn : Dn = AQ(n+1)' (AQ Q AQ) AQn OE-! Q Q = n Q
which maps a0 . . .an 7! a0 . . .an-2 an-1.jR(an) and one checks that OE.is *
*an isomorphism
of cosimplicial groups and hence of cochain complexes.
We have H.(D.) = H0(D.) = Q, a contracting homotopy being given by
(4) Hn : Dn = AQ(n+1)-! Dn-1 = AQn ; a0 . . .an 7! o(a0)a1 . . .an.
Define a subcomplex . D.by
4
Beta-elements and divided congruences
Xn Xn
n := @i(Dn-1) = AQi Q AQ(n-i) Dn = AQ(n+1),
i=1 i=1
for n > 1 and 0 := 0. One checks that the composition
__n n n (OEn)-1n n n
'n : ,! ,! Q -! D -! D =
isPinjective for all n > 0. This is obvious for n = 0, and for n > 1 it follows*
* from (\n-1i=0ker(oei)) \
( ni=1im(@i)) = 0, which in turn is an easy consequence of thePcosimplicial id*
*entities: One shows
by descending induction on 1 6 j 6 n that (\n-1i=0ker(oei))\( ni=jim(@i)) = 0.*
* Observe that Dn= n
is isomorphic to the group labelled E* Gn*in [L1, p. 404] for A = BP.
We define a cochain complex Q.by the exactness of
__.'. . . ss. .
(5) 0 -! -! D = -! Q -! 0,
__ An
hence Qn ' AQ(n+1)= n + im( ).
From the definitions of the differential of D.and n one obtains (BQn Qn deno*
*ting the boundaries)
__ An
Qn=BQn ' AQ(n+1)=" n+ im( ),
P n+1 (i-1) (n+1-i)
where " n:= n + Q AQn = i=1AQ Q AQ .
The alternative description of Extwe are aiming for is the following.
Proposition 3. For any n > 1, the connecting homomorphism ffi of (5) is an isom*
*orphism
__. n+1
Hn(Q.) -ffi!Hn+1( ) = Ext .
Proof.One readily sees that the contracting homotopy (4) of D. respects the sub*
*complex . in __
positive dimensions, hence the middle term of (5) is acyclic in these dimension*
*s. |__|
To explicitly compute ffi, it is useful to note that the differential of D.= .h*
*as the simple form
Dn= n -d!Dn+1= n+1 , [a0 . . .an] 7! [1 a0 . . .an],
as is immediate from the definitions. To compute ffi-1, we consider the zig-zag
oooo_ _'n+1//_ _Hn+1//_ __n+1n+1n+1nnssn////_n////_nn
Extn+1__________________________________Z(DD =)=Q Q =BQOO
____________ffi-1______________________________|
_______________ |
___________________________O|
_________--_____________*
*______Hn(Q.).?
5
Jens Hornbostel and Niko Naumann
One checks that the dotted arrow exists and is the inverse of ffi.
Now let p be a prime, r > 1 an integer, and let ffi0: Extn(A, A=pr) -! Extn+1be*
* the connecting
r
homomorphism associated to the short exact sequence of -comodules 0 -! A .p-!A*
* -! A=pr -!
0. Consider the diagram
__n __n __n+1ff
(6) ffi0: Extn(A, A=pr)oZ(ooo_A A=pr)oooo{z_2 |dz 2 pr _}___//Extn+1OO
________ |
|| ________________'ffi|
fflffl| ___((_____|
Qn=BQn oo_______?Hn(Q.).`_
__n+1
Here, ff(z) := [y] for any y 2 satisfying dz = pry and (z) := ssn(p-r'n(z*
*)) mod BQn. The
upper horizontal line is ffi0 by definition and one checks that factors throu*
*gh the dotted arrow
and makes the diagram commutative. Hence, when displaying an element of im(ffi0*
*) in H.(Q.) rather
than in the usual cobar complex, one does not have to compute the cobar differe*
*ntial implicit in ff
but only the (easier) map .
Finally, assume that everything in sight is graded where the grading on Dn, n, *
*etc. is by total
degree. For a fixed k 2 Z, consider the commutative diagram
__ An n `. ffi
Qn=BQn ~=AQ(n+1)=" n+ im( )oo___H?(Q ) __'__//_Extn+1
OO OO| OO|
j|| || ||
?O| __ An ` ?O| ' O|?
AQ(n+1),k=" n,(k)+ im( o)ko__?Hn,k(Q.)___//__Extn+1,k.
Here " n,(k):= " n\ AQ(n+1),k. One checks that j is well defined and injective,*
* and Hn,k(Q.) is
defined to be the pull back of Hn(Q.) along j. The commutative diagram induces *
*an isomorphism
Hn,k(Q.) -'!Extn+1,kas indicated.
For example, for (A, ) = (BP *, BP*BP ) and n = 1 we obtain an inclusion
(BP Q BPQ)(k)
(7) Ext2,k _____________________________
BP kBP + (BP Q Q + Q BPQ)(k)
which is important for us since the f-invariant is defined in terms of the grou*
*p on the right hand
side.
To effectively compute representatives of fi-elements in the complex Q.one proc*
*eeds as follows. Let
t, s, r > 1 be integers as in section 2.1 and ffi, ffi0the coboundary maps intr*
*oduced there. Fix k 2 Z
and x 2 H0,k(A=(pr, vs1)), that is
x 2 C0,k(A=(pr, vs1)) = (A=(pr, vs1))k
is an invariant element (C indicates the reduced cobar complex). As ffi is the*
* connecting homomor-
phism determined by the short exact sequence of complexes obtained by applying *
*C to (2), we
compute ffi(x) as follows: Lift x to y 2 (A=(pr))k and compute the cobar differ*
*ential
6
Beta-elements and divided congruences
__k __ r k 1,k r
d = jR - jL : C0,k(A=pr) = (A=pr)k -! (A=pr A ) = ( =p ) = C (A=p )
__ __k
obtaining d(y) 2 ( =pr)k = =pr. Note that this computation requires knowledge*
* of jR(y) mod pr.
Now d(y) will be divisible by vs1, hence
__ r k
d(y) = vs1z in( =p )
__ 1,k-2s(p-1) *
* __k-2s(p-1)
with z 2 ( =pr)k-2s(p-1)= C (A=pr) representing ffi(x). Lift z to some w*
* 2 . To
proceed, we use diagram (6): w lies in the last but one group of the top row, h*
*ence we compute
(w) 2 H2(Q.) which is our representative for ffi0(ffi(x)) 2 Ext2. This require*
*s to compute OE-1. Ob-
serve for example that OE-1(t1) = 1_v1-v1_1p.
2.3 Elliptic homology theories and divided congruences: the f-invariant
We refer the reader to [L1] or [HBJ ] for the notion of elliptic homology with *
*respect to the congruence
subgroup 1(N). In this section, E denotes the spectrum associated to the homol*
*ogy theory with
coefficient ring E* = M*(Z(p), 1(N)), see section 3.2 for the notation. Finall*
*y, ff : BP -! E denotes
the orientation.
By the naturality of the constructions in section 2.2 we have a commutative dia*
*gram for any k > 0
" ff 2,k
Ext2,k[BPO]_____________//"E`xt[E]" `
| |
(7)| (7')|
fflffl| fflffl|
_____(BPQ_BPQ)(k)____ff_ff//__(EQ_EQ)(k)__
BPkBP+(BPQ Q+Q BPQ)(k) EkE+(EQ Q+Q EQ)(k).
The injectivity of ff holds for any Landweber exact theory E of height at least*
* two, [L2, Proof
of 4.3.2]. To proceed, however, we will use a more subtle property of E, namely*
* the topological
q-expansion principle. We put
Xk Mk
D__k:= {f = fi2 EQ,2i|there areg0 2 Q, gk 2 EQ,2ksuch that(f + g0+ gk)(q*
*) 2 Z(p)[[q]]}
i=0 i=0
where f(q) denotes the q-expansion of f at the cusp infinity, and for = 1(N)*
* we set Z :=
Z[_1_N, iN ] if N > 1 and ZSL2(Z):= Z[1_6] as in [L1]. We then define
(EQ EQ)(2k) X X 0
(8) '2 : __________________________-! D__ Q=Z , fi gj 7! -q (fi)gj,
E2kE + (EQ Q + Q EQ)(2k) k i+j=k i+j=k
where q0(f) is the constant term of the q-expansion of f at the cusp infinity. *
*The composition '2O '
is injective [L1, Proposition 3.9] and hence so is the f-invariant
f : Ext2,2k[BP ] ,! D__k Q=Z .
7
Jens Hornbostel and Niko Naumann
We remark that our grading of E* is the topological one, i.e. elements of dimen*
*sion 2k correspond
to modular forms of weight k.
G. Laures describes Ext2using the canonical Adams resolution [R , Definition 2.*
*2.10] instead of the
cobar resolution.
Proposition 4. For k > 0 even, the above definition of the f-invariant Ext2,k-!*
* D__k=2 Q=Z
coincides with the one given in [L1].
(k)
Proof.We have to show that the map Ext2,k,! _____(BPQ_BPQ)_______BPkBP+(BPQwQ+Q*
*eBPQ)(k)constructed in section
2.2,(7) coincides with the one of [L1]. We know [R , Lemma A.1.2.9 (b)] that, u*
*p to chain homo-
topy, there is a unique map from the unreduced cobar resolution A A.to the*
* canonical Adams
resolution BP*(BP ^ B~P^.) where B~P! S0 j!BP !d B~Pis an exact triangle in th*
*e stable homo-
*
* ss*(idBP^2^d^n)
topy category, see also [Br, Lemma 3.7] Now one checks that A(n+1)~=ss*(BP ^n*
*+2) !
ss*(BP ^2^ ( B~P)^n) is a map of chain complexes where the isomorphism follows *
*from [R , Lemma
2.2.7] and induction. So the claim reduces to the fact that the triangle
ss*(idBP^2^d)
Z2,k( A.)_____________________________//Z2,k(ss*(BP ^2^ ( B~P)^.))
RR iii
RRRR iiiiii
RRRR iiiii
RR))R ttiii
_____(BPQ_BPQ)(k)____
BPkBP+(BPQ Q+Q BPQ)(k)
commutes. This follows using that our maps o, H2 and ss*(d ^ d) correspond to r*
*, ae and the isomor-
phism D1= "1~=G2 = ss*(( B~P)^2) Q in [L1, section 3.1] (where we define o by*
* mapping_all vito
zero). *
* |__|
Note the degree shift and the factor 2 for the f-invariant sss2k-! D__k+1 Q=Z *
*(both are missing in
[L1, p. 411]).
3. Arithmetic computations
In section 3.1, we review results of N. Katz on divided congruences and establi*
*sh a relation between
BP -theory and the mod p Igusa tower (Theorem 5). In section 3.2 we give explic*
*it computations
for elliptic homology of level 3 and the corresponding divided congruences.
3.1 Divided congruences
We review parts of [K1 ]. Some technical remarks are in order: In loc. cit. N. *
*Katz works with level-N
structures of fixed determinant for N > 3. To confirm with general policy in al*
*gebraic topology we
wish to consider 1(N)-structures instead, which are representable only for N >*
* 5. More seriously,
one has to check that the relevant part of [K1 ] works for this different modul*
*i problem. This we
did, but we will not explain the details here and only remark that both the geo*
*metric irreducibility
of the moduli spaces and the irreducibility of the Igusa tower are valid for 1*
*(N). Furthermore, we
will use these results for p > 5 and N = 1 and for p = 2 and N = 3. These cases*
* can be handled by
using auxiliary rigid level structures and taking invariants under suitable fin*
*ite groups_as in [K1 ].
Fix a level N > 5, a prime p not dividing N and a primitive N-th root of unity *
*i 2 Fp. We
8
Beta-elements and divided congruences
put k := Fp(i), W := W (k) (Witt vectors) and also denote by i 2 W the Teichm"u*
*ller lift of i.
Finally, K denotes the field of fractions of W and for any Z(p)-algebra R we de*
*note by Mk(R, 1(N))
the R-module of holomorphic modular forms for 1(N) of weight k and defined ove*
*r R , see e.g.
[K3 ] or [L1]. The 1(N) is omitted from the notation if it is clear from the c*
*ontext. We fix a lift
Ep-12 Mp-1(W ) of the Hasse invariant. The existence of such a lift puts furthe*
*r restrictions on p
and N which are satisfied in our applications.
We define the ring of divided congruences D by
M*(W ) D := {f 2 M*(K)|f(q) 2 W [[q]]} M*(K),
where f(q) is the q-expansion of f at the cusp infinity. For n > 0 we also defi*
*ne
_ n !
M
Dn := D \ Mi(K) D and
i=0
Mn
D__n:= Dn + K + Mn(K) Mi(K)
i=0
which is consistent with the definition of the previous section. The group D__k*
*considered in [L1]
differs from the D__kabove because the ring of holomorphic modular forms has be*
*en localised in [L1].
This difference is not serious because the f-invariant factors through holomorp*
*hic modular forms
[L1, Proposition 3.13].
The ring D carries a uniformly continuous Z*p-action (the diamond operators) de*
*fined by
X X
[ff]( fi) := ffifi,
i i
where ff 2 Z*pand fi2 Mi(K). We put 0 := Z*pand n := 1+pnZp for n > 1 and V1,*
*n:= (D=pD) n
for n > 0. Then
V1,0 V1,1 . . .D=pD
is an ind-'etale Z*p-Galois extension, the mod p Igusa tower. So, V1,0 V1,1is *
*a (Z=p)*-Galois
extension and for all n > 2 V1,n-1 V1,nis an 'etale Z=p-extension and hence an*
* Artin-Schreier ex-
tension. An immediate computation with diamond operators shows that the composi*
*tion M*(W ) !
D ! D=pD factors through V1,1 D=pD. It is a result of P. Swinnerton-Dyer that *
*this induces an
isomorphism
(9) M*(W )=(p, Ep-1- 1) -'!V1,1,
see [K1 , Corollary 2.2.8].
By Artin-Schreier theory, given n > 2 and x 2 D=pD satisfying [ff](x) = x for a*
*ll ff 2 n and
[1 + pn-1](x) = x + 1 one has V1,n= V1,n-1[x] and the minimal polynomial of x o*
*ver V1,n-1is
T p- T - a for some a 2 V1,n-1.
9
Jens Hornbostel and Niko Naumann
At this point we can establish a first relation between BP-theory and divided c*
*ongruences. Consider
the ring extensions
M*(W ) D W [[q]].
We have a formal group F over M*(W ) induced by the universal elliptic curve. T*
*he base change of
F to W [[q]] is the formal completion of a Tate elliptic curve and is thus isom*
*orphic to dGm. Implicit
in [K1 ] is the fact that D is the minimal extension of M*(W ) over which F bec*
*omes isomorphic
to dGm, i.e. D is obtained from M*(W ) by adjoining the coefficients of an isom*
*orphism F ' dGm
defined over W [[q]]. This is what underlies N. Katz' construction [K1 , Sectio*
*n 5] of a sequence of
elements dn 2 D which modulo p constitute a sequence of Artin-Schreier generato*
*rs for the mod p
Igusa tower.
Since the elements tn 2 BP *BP are the coefficients of the universal isomorphis*
*m of a p-typical
formal group law, one may expect a relation between the tn and the dn. To formu*
*late this, denote
by ff : BP* -! M*(W ) the classifying map of F and consider the composition
(3) 2ff ff 2 -q0 id
OE : BP*BP BP*BP Q ' BP Q -! M*(K) -! M*(K).
Note that the map '2 in (8) composed with the orientation ff is a quotient of O*
*E.
The topological_q-expansion principle guarantees that Tn := OE(tn) 2 D for n > *
*1 and we can thus
define Tn := (Tn mod pD) 2 D=pD.
__ __ __ *
* __
Theorem_5. For any n > 1 we have [1 + pk](T n) = Tn for k > n and [1 + pn](T n)*
* = Tn+ 1.
Hence Tn is an Artin-Schreier generator for the extension V1,n V1,n+1.
P
Proof.Let ! = ( n>1antn-1)dt be the expansion along infinity of a normalised (*
*i.e. a1 = 1)
invariant differentialPon the universal elliptic curve. Then an 2 Mn-1(W ) and *
*the logarithm of
the p-typification is n>0apn_pntpn 2 M*(K)[[t]], i.e. the classifying map ff *
*: BP* -! M*(W ), when
tensored with Q, sends ln 2 BP Q,2(pn-1)to apn_pn2 Mpn-1(K), see [R , Theorem A*
*.2.1.27] for the
definition of the ln (=~n in the notation of loc. cit.).
Defining d0 := 1 and dn (n > 1) recursively by
Xn dpi a n
(10) _n-i_i= _p_n,
i=0 p p
_
N. Katz shows in [K1 , Corollary 5.7] that the dn:= (dn mod p) 2 D=pD behave un*
*der the diamond
__ P n pi
operators as claimed for the Tn. In BP *BP Q we have jR(ln) = i=0litn-i and *
*we apply OE to
this relation to obtain
apn_ Xn q0(api) pi
= ______Tn-i
pn i=0 pi
which, using (10), implies
10
Beta-elements and divided congruences
Xn dpi Xn q0(a i) i
(11) _n-i_i= ____p_iTnp-i.
i=0 p i=0 p
We now proceed by induction on n > 1. For n = 1 we have d1 + 1=p = q0(a1)T1 + q*
*0(ap)=p. Also,
q0(a1) = 1 since a1 = 1 and q0(ap) 2 1 + pW because_ap_reduces mod p to the Has*
*se invariant
which has q-expansion equal to 1. We obtain T1 = d1+ ff for some ff 2 k. As ff *
*is invariant under
all diamond operators, our claim for n = 1 is obvious.
Assume that n > 2. From (11) and a1 = 1 we obtain
Xn dpi Xn q0(a i) i
Tn = _n-i_i- ____p_iTnp-i.
i=0 p i=1 p
For k > n we know that the terms involving di are invariant mod p under [1 + pk*
*] whereas the
remaining terms are likewise invariant by the induction hypothesis.
Finally, we have
Xn dpi Xn q0(a i) *
* i
[1 + pn]Tn = [1 + pn]dn + [1 + pn]( _n-i_i) - [1 + pn]( ____p_iT*
*np-i).
i=1 p i=1 p
Here we have_[1 +_pn]dn dn + 1 (pD) and the remaining terms are invariant. Th*
*us, indeed,_
[1 + pn](T n) = Tn+ 1. *
* |__|
3.2 Modular forms
For a prime p > 5, the following is well known [L1, Appendix]:
M*(Z(p), 1(1)) = Z(p)[E4, E6],
where E4 and E6 are the Eisenstein series of level one of the indicated weight.*
* For the discriminant
, the ring of meromorphic modular forms is given by Z(p)[E4, E6, -1] and the *
*usual orientation
BP*- ! Z(p)[E4, E6, -1]
is Landweber exact of height 2 and factors through Z(p)[E4, E6]. A similar resu*
*lt holds for p > 3
and M*(Z(p), 1(2)) = Z(p)[ffi, ffl].
The purpose of this section is to give analogous results for 1(3) and p = 2, c*
*.f. [St] for related
results.
Consider the elliptic curve
E : y2 + a1xy + a3y = x3
defined over R := Z[1=3][a1, a3, -1] where = a33(a31- 27a3) is the discrimin*
*ant of the given
Weierstrass equation. Note that, unlike in level one, is not irreducible as a*
* polynomial in a1 and
11
Jens Hornbostel and Niko Naumann
a3 and we put f := a3, g := a31- 27a3, hence = f3g.
The section P := (0, 0) 2 E(R) is of exact order 3 in every geometric fibre as *
*follows from [Si1,
III,2.3] and ! := dx=(2y + a1x + a3) is an invariant differential on E.
The following may be compared with [St, Lemma 11]:
Proposition 6. The above tuple (E=R, !, P ) is the universal example of an elli*
*ptic curve over a
Z[1=3]-scheme together with a point of order 3 and a non-zero invariant differe*
*ntial.
Proof.We have to show that whenever T is a Z[1=3]-scheme and E0=T is an ellipti*
*c curve with
non-zero invariant differential !0and P 02 E0(T ) of exact order 3, there is a *
*unique map OE : T -!
Spec(R) such that OE*(E, P, !) = (E0, P 0, !0). We show the uniqueness of OE fi*
*rst. This amounts to
seeing that the only change of coordinates
x = u2x0+ r , y = u3y0+ u2sx0+ t
with r, s, t 2 R and u 2 R* (see [Si1, III Table 1.2]) preserving (E, P, !) is *
*the identity, i.e. r = s =
t = 0 and u = 1.
From x0(P ) = y0(P ) = 0 we obtain r = t = 0. Next, a4 = a04implies -sa3 = 0, h*
*ence s = 0 because
= f3g and thus also f = a3 is a unit in R. Finally, !0= u! forces u = 1.
Given the uniqueness of OE in general, its existence is a local problem on T an*
*d we can assume that
T = Spec(S) is affine and E0=T is given by a Weierstrass equation with coeffici*
*ents a0i2 S. Moving
P 0to (0, 0) gives a06= 0. We claim that a032 S*: This can be checked on geomet*
*ric fibres where
it follows from [Si1, III,2.3] and the fact that (0, 0) has order 3 (if a03vani*
*shed on some geometric
fibre the point (0, 0) would have order 2 in that fibre). Using this, one finds*
* a transformation such
that (dy)P0 = 0 in E0=T,P0, hence a02= a04= 0. We thus have some _ : T -! Spec*
*(R) such that
_*(E, P ) = (E0, P 0) and _*(!) = u!0for some u 2 S*. Adjusting _ using u, i.e.*
* multiplying_the a0i
by u-i, we obtain the desired OE. *
* |__|
We conclude that the ring of meromorphic modular forms is given as
Mmer*(Z(2), 1(3)) = Z(2)[a1, a3, -1]
with aiof weight i, and likewise for any other prime different from 3 in place *
*of 2.
As usual, t = -x=y is a local parameter at infinity for E=R which is normalised*
* for ! and hence
determines a 2-typical formal group law over Mmer*(Z(2), 1(3)). Using [Si1, p.*
* 113] one checks that
the corresponding classifying map
ff : BP* -! Mmer*(Z(2), 1(3))
satisfies ff(v1) = a1and ff(v2) = a3for the Hazewinkel generators vi. Thus ff m*
*akes Mmer*(Z(2), 1(3))
a Landweber exact BP algebra of height 2.
Using the orders of f and g at the two cusps 0 and 1 of X1(3), one can check th*
*at the ring of
holomorphic modular forms is given by
(12) M*(Z(2), 1(3)) = Z(2)[a1, a3].
12
Beta-elements and divided congruences
We stick to the notations of section 3.1 for p = 2 and N = 3. For example, i de*
*notes a primitive cube-
root of unity and W = W (F2(i)) = W (F4) = Z2[i] is the unique unramified quadr*
*atic extension of
Z2.
To study divided congruences we will need to know the q-expansions of a1 and a3*
*. Given a Dirichlet
character O, we consider it as a function on Z as usual and define for k > 0 an*
*d n > 1
X
oeOk(n) := O(d)dk.
16d|n
In the following, O will always denote the unique non-trivial character mod 3
O : (Z=3Z)* -! C*.
Proposition 7. The q-expansions of a1 and a3 at the cusp infinity are given as *
*follows.
X O
a1(q) = (1 + 2i)(1 + 6 oe0(n)qn) and
n>1
1 X O n
a3(q) = (1 + 2i)(-_ + oe2(n)q ) inW [[q]].
9 n>1
Proof.From (12) we know that rkM1(Z(2)) = 1 and rkM3(Z(2)) = 2. Using [K2 , sec*
*tion 2.1.1] we
see that
X O
(13) 6G1,O(q) = 1 + 6 oe0(n)qn 2 M1(Z(2)) and
n>1
1 X O n
G3,O(q) = -_ + oe2(n)q 2 M3(Z(2)).
9 n>1
We have evaluated L(0, O) = 1=3 and L(-2, O) = -2=9 using [Ne, Theorem VII.2.9]*
* and [Wa ,
formula following Proposition 4.1 and Exercise 4.2(b)].
It is easy to see that G3,O(0) = 0, i.e. G3,Ovanishes at the cusp 0. Below, we *
*explain how to compute
the following values of a1 and a3 at the cusps zero and infinity.
(14) a1(1) = 1 + 2i
1
(15) a3(1) = -_ (1 + 2i)
9
(16) a3(0) = 0.
Using these values and the dimensions of the spaces of modular forms of weight *
*1 and 3, we conclude
that a1 = 6(1 + 2i)G1,Oand a3 = (1 + 2i)G3,O, hence that a1 and a3 have desired*
* q-expansions by
(13). We are using the fact that the map M3(Z(2)) C = M3(C) -! C2, f 7! (f(1)*
*, f(0)) is an
isomorphism, as follows from the theory of Eisenstein series.
To establish (14) and (15) one has to evaluate a1 and a3 at the tuple (T (q), !*
*can, P ) consist-
ing of the Tate curve T (q)=Z((q)), its canonical invariant differential !can a*
*nd a specific sec-
tion P 2 T (q)(Z[i]((q)))[3]. To do so, one may use J. Tate's uniformisation [*
*Si2, p. 426] to
13
Jens Hornbostel and Niko Naumann
write T (q)=(Z[[q]]=(q3)) in Weierstrass form, the point P having coordinates (*
*X(q, i), Y (q, i)). One
then uses Weierstrass transformations to bring (T (q), !can, P )=Z[[q]]=(q3) to*
* the standard form of
Proposition 6. The coefficients a1 and a3 of the Weierstrass equation thus obta*
*ined are by defi-
nition a1(1) and a3(1). The computation for (16) is similar, the point P has to*
* be replaced_by
Q = (X(q, q1=3), Y (q, q1=3)). *
* |__|
Remark 8. In E. Hecke's notation [He], we have a1 = 9i_ssG1(o, 0, 1, 3) and a3 *
*= 27i_4ss3G3(o, 0, 1, 3).
Note that a1(q) 1 mod 2, hence a1 2 M1(Z(2), 1(3)) is a lift of the Hasse in*
*variant for p = 2.
From section 3.1 we know that
(9)
V1,0= V1,1' M*(W, 1(3))=(2, a1- 1) = k[a3]
0(a1)-a1 __
(k := W=2W = F4) and that, for T := q______22 D, T := (T mod 2) 2 D=2D is an Ar*
*tin-Schreier
__2 __
generator for V1,1 V1,2, in particular T + T 2 V1,1' k[a3] and for later use *
*we will need the
following more precise result.
__2 __
Proposition 9. T + T = 1 + a3.
Proof.Recall that the q-expansionPmap V1,1 D=pDP,! k[[q]] is injective [K1 , (*
*1.4.6) for m = 1].
In k[[q]] we have T = n>1oeO0(n)qn and T 2= n>1oeO0(n)q2n, hence
X O O
T 2+ T = (oe0(n=2) + oe0(n))qn,
n>1
where we understand that oeO0(n=2) = 0 for n odd. To complete the proof, one ne*
*eds to check that
for all n > 1 one has
oeO0(n=2) + oeO0(n) oeO2(n) mod 2,
*
* __
and we leave this exercise in elementary number theory to the reader. *
* |__|
4.f-invariants and Kervaire invariant one
In this section, we compute the f-invariants of two infinite families of fi-ele*
*ments including the
Kervaire elements fi2n,2nand explain the relation of our results with the Kerva*
*ire invariant one
problem.
4.1 f(fit) for t not divisible by p
Fix a prime p and the level N as N = 1 for p > 5, N = 5 for p = 3 and N = 3 for*
* p = 2. We keep the
notations of section 3.1 for this choice2of p and N. Given an integer t > 1 not*
* divisible by p, recall
that fit= ffi0ffi(vt2) 2 Ext2,2t(p -1)-2(p-1)[BP ] has its f-invariant in D__n *
* Q=Z, n := t(p2- 1) - (p - 1).
When trying to express f(fit) in terms of divided congruences, we encounter wha*
*t is in fact the
major obstacle at the moment for using the arithmetic of divided congruences in*
* homotopy theory:
The group D__n Q=Z is not directly related to D. Instead, we have D__n= D+K +Mn*
*(K) by definition
and there is a canonical surjection
14
Beta-elements and divided congruences
_ n ! _ !
M Mn
ss : Dn Q=Z ' Mi(K) =Dn -! D__n Q=Z ' Mi(K) =D__n
i=0 i=0
which is split because its kernel is divisible, hence W -injective. In particul*
*ar, ss remains surjective
when restricted to p-torsion
(17) ss : Dn Q=Z[p] -! D__n Q=Z[p],
note that f(fit) 2 D__n Q=Z[p]. The group Dn Q=Z[p] is related to the ring of*
* divided congruences
as follows:
(18) _ : Dn Q=Z[p] -'!Dn=pDn ,! D=pD,
where the first arrow is multiplication by p and the injectivity of the last ma*
*p is immediate. What
we will do is to compute some element in D=pD in the image of _ which under ss *
*projects to f(fit).
At the low risk of confusion we will continue to label such an element, which i*
*s in general not
unique, as f(fit). Recall that we have fixed an elliptic orientation ff : BP* -*
*! M*(W ) and denote
0(ff(v1))
T := ff(v1)-q___p2 D=pD. We also put b := ((q0(ff(v2)) mod p) 2 k.
Theorem 10. For an integer t > 1 not divisible by the fixed prime p, we have
f(fit) = bt- (T p- T + b)t2 V1,0 D=pD.
Proof.Note first that from section 3.1 we know that T is an Artin-Schreier gene*
*rator for V1,1 V1,2,
hence T p- T 2 V1,1. A short computation with diamond operators, which we leave*
* to the reader,
shows that in fact T p- T 2 V1,0, hence also bt- (T p- T + b)t2 V1,0.
We introduce a := q0(OE(v1)) and compute as explained at the end of section 2.2*
* using the notations
introduced there. From jRv2 v2+ v1tp1- vp1t1 mod p we obtain
Xt `t'
jRvt2 vt2+ vt-i2vi1ti1(tp-11- vp-11)imod p,
i=1 i
hence
Xt `t'
w = vt-i2vi-11ti1(tp-11- vp-11)i
i=1 i
and
` ' ` ' i
1X t t t-i i-1 1 v1- v1 1
(w) = _ (v2 v1 1) _____________ .
p i=1 i p
_` ' !i
1___v1-_v1__1_p-1 p-1 (BP Q BPQ)(2n)
- v1 1 2 _______________________________.
p BP 2nBP + (BP Q Q + Q BPQ)(2n)
As in section 2.3 we apply '2O (ff ff) to this expression to obtain, denoting*
* ff(v1) 2 Mp-1(W ) as
v1 for simplicity,
15
Jens Hornbostel and Niko Naumann
` ' ` 'i_` ' p-1 !i
1 Xt t t-ii-1 v1- a v1- a p-1
-_ b a _____ _____ - a =
p i=1 i p p
" _ ` ' _` ' ! !t#
1 t v1- a v1- a p-1 p-1
-__ -b + a _____ _____ - a + b =
pa p p
__ ` ' _` ' ! !t ! n
-1_ v1- a v1- a p-1 p-1 t M
a _____ _____ - a + b - b 2 Mi(K).
pa p p i=0
This is a representative for f(fit) in Dn Q=Z[p] to which we have to apply th*
*e map _ from (18)
to obtain an element in D=pD. For this, note that a 1 mod p because v1 reduce*
*s to the Hasse
invariant mod p. This allows us to put a = 1 in the above expression (but not t*
*o replace v1-a_pby
v1-1_ 2
p ; this would require the congruence a 1 mod p , which does not hold in ge*
*neral). We then
obtain indeed
-((T (T p-1- 1) + b)t- bt) = bt- (T p- T + b)t.
*
* __
*
*|__|
Remark 11. Assume that p > 5 in the situation of Theorem 10. In general, the el*
*liptic orientation
will not map v1 to the Eisenstein series Ep-1 of weight p - 1 and level one. Bu*
*t ff(v1) and Ep-1
can only differ by a modular form divisible by p and we may thus change the ori*
*entation to force
ff(v1) = Ep-1. Assuming this, we see that f(fi1,1,1) = Ep-1-1_p2- 1_p(Ep-1-1_p)*
*p, as first computed by G.
Laures [L1, p. 414] (where the second summand is missing).
Remark 12. The injectivity of the f-invariant together with the known structure*
* of Ext2[BP ] pro-
vides some non-trivial information about the arithmetic of divided congruences *
*as follows. Fix some
x 2 Ext2,k[BP ] of order pr. Then f(x) 2 D__k Q=Z will be of order pr, hence a*
* representative of
f(x) in Dk Q=Z will be of order ps for some s > r. Thus the f-invariant relat*
*es the order of a
fi-element to the (non-)existence of a certain divided congruence.
Let us assume that r = 1 as is the case for all fit,s,rconsidered in this artic*
*le. Then our results
show that our representatives in D Q=Z have order p and the non-trivial addit*
*ional information
on divided congruences is then that they do not lie in the kernel of D Q=Z -!*
* D__ Q=Z.
To give an example, assume that we are in the situation of Remark 11. The arith*
*metic of dividedLcon-
gruences shows that F := Ep-1-1_p2- 1_p(Ep-1-1_p)p 2 Dp(p-1) Q=Z is of order p,*
* i.e. F 2 p(p-1)i=0Mi(K)
has a q-expansion with denominator exactly p. The additional information is the*
*n that for any ff 2 K
and f 2 Mp(p-1)(K) the q-expansion of F + ff + f will still have exact denomina*
*tor p.
Example 13. Fix p = 5 and set g2 := _1_12E4 and g3 := -1_216E6 as in [K1 ]. The*
* comparison of the
logarithm of the universal p-typical formal group law [R ] and the correspondin*
*g coefficients of the
logarithm of the elliptic curve (E, !) [K1 , (5.0.3)] (p-typification does not *
*change these coefficients)
ap2-ap+1p
shows that the orientation a maps v1 to ap and v2 to _______p, the ai denoting *
*the normalised
*
* 6
(multiplied with -1=2) aiof [K1 , p. 351]. One deduces that v1 maps to -8g2 and*
* v2 maps to a25-a5_5.
A computation with Maple shows that a25= 129761280g32g23+ 32440320g43+ 3784704g*
*62(and also
that the correct value for the unnormalised a11 is -2520g2g3 and not -512g2g3).*
* It follows that
q0(v1) = -2_3and q0(v2) = -4900_310, so Theorem 10 may be rephrased in terms of*
* the Eisenstein series
g2 and g3.
16
Beta-elements and divided congruences
4.2 Projecting to the Kervaire element
In this section, we compute f(fis2n,2n) for n > 0 and s > 1 odd at the prime p *
*= 2. Using this, we
are able to determine a single coefficient in the f-invariant of a (U, fr)2- ma*
*nifold of dimension 2n
the non-vanishing of which is necessary and sufficient for the corner of X to b*
*e a Kervaire manifold,
that is having Kervaire invariant one. See [L2] for the notion of cobordism of *
*manifolds with corners.
We begin by recalling the well-known relation of the Kervaire invariant to cert*
*ain fi-elements, due
to W. Browder [B ]. Fix some n > 3. We have a homomorphism
K : sss2n-2-! Z=2
which sends the class of a stably framed manifold to its Kervaire invariant. Co*
*nsider on the other
hand the composition
n 2,2n 2
K0: sss2n-2-! sss2n-2[2] -! E2,21[HZ=2] ,! E2 [HZ=2] = Z=2 . hn-1.
Here, the first map is the projection to the 2-primary part, the second is the *
*projection onto F 2=F 3
in the (classical) Adams spectral sequence at p = 2, the third is an edge homom*
*orphism and the
final equality is due to J. Adams, [R , 3.4.1, c)].
Proposition 14. K = K0.
Proof.For any y 2 sss2n-2[2] we have K(y) = 1 if and only if y has Adams filtra*
*tion 2. This_is_
implicit in [B ], c.f. [BJM2 , p. 144]. *
* |__|
We can easily obtain a similar homotopy theoretic description of K using BP ins*
*tead of HZ=2.
n
Proposition 15. Let n > 2. Then Ext2,2is a direct sum of cyclic groups of order*
* 2. It is generated
by the element ff1.ff2n-1-1and the elements fis2i,2iwith s odd and i > 0 such t*
*hat (3s-1)2i+1= 2n
and the case (s, i) = (1, 0) has to be omitted.
Proof.This follows from [R , Corollary 5.4.5]. Observe that the __fftin loc. ci*
*t equals fft_as t is odd,
see [Sh, Theorem 1.5] or [R , Theorem 5.2.6]. *
* |__|
n
Remark 16. The Lemma shows that the number of generators of Ext2,2is [n=2] + 1 *
*for n > 3.
The low dimensional cases are as follows.
Ext2,4: ff21
Ext2,8: ff1ff3, fi2,2
Ext2,16: ff1ff7, fi4,4, fi3,1
Ext2,32: ff1ff15, fi8,8, fi6,2
Ext2,64: ff1ff31, fi16,16, fi12,4, fi11,1
Ext2,128: ff1ff63, fi32,32, fi24,8, fi22,2
Ext2,256: ff1ff127, fi64,64, fi48,16, fi44,4, fi43,1.
17
Jens Hornbostel and Niko Naumann
Now we consider the composition
n 2,2n
K00: sss2n-2-! sss2n-2[2] -! E2,21[BP ] ,! E2 [BP ] -! Z=2 . fi2n-2*
*,2n-2
which is defined in analogy with K0, the final map being the projection to the *
*Z=2-summand
generated by fi2n-2,2n-2.
Proposition 17. K00= K.
Proof.We have the Thom reduction : Ext*[BP ] -! Ext*[HZ=2] which satisfies (*
*fi2n-2,2n-2) =
n
h2n-1and is zero on all other generators of Ext2,2[BP ], see [R , 5.4.6, a)] an*
*d Proposition_15. The
result then follows from Proposition 14. *
* |__|
Let X be a (U, fr)2-manifold of dimension 2n. From the above, we see that the c*
*orner of X is
a Kervaire manifold if and only if the f-invariant of X contains fi2n,2nas a su*
*mmand. Thus, one
certainly wants a more geometric description of the f-invariant (or just its pr*
*ojection to fi2n-2,2n-2).
In principle, it is possible to obtain such a description in terms of Chern num*
*bers of X, simply
because they determine the (U, fr)2-bordism class of X [L2], but the necessary *
*computations become
quite complicated already in low dimensions. At the end of this section, we wil*
*l explain how divided
congruences might simplify such computations. We then would like to generalise *
*Theorem 18 below
to higher dimensions.
Recall [L2, section 4.1] that if X is a (U, fr)2-manifold then there is a decom*
*position of its stable
tangent bundle T X = T X(0) T X(1)and we have Chern classes c(j)i2 H2i(X, Z) a*
*ccordingly
(i > 0, j = 0, 1).
Theorem 18. a) Let X be a (U, fr)2-manifold of dimension 4 and put q :=< c(0)1c*
*(1)1, [X] >2 Z.
Then q is odd if and only if the corner of X has Kervaire invariant 1. If q is *
*even, then the corner
of X is the boundary of a framed manifold.
b) Let X be a (U, fr)2-manifold of dimension 8 and put q :=< c(0)1(c(1)31+ c(1)*
*1c(1)2+ c(1)3) + (c(0)2+
c(0)21)(c(1)2+ c(1)21), [X] >2 Z. Then q is odd if and only if the corner of X *
*has Kervaire invariant 1.
If q is even, then the corner of X is the boundary of a framed manifold.
P i
Proof.If i>0lix2 is the logarithm of the universal 2-typical formal group law*
*, then (see [L2,
Example 4.2.4])
exp(x) = x - l1x2+ 2l21x3- (5l31+ l2)x4 (mod x5)
and thus
x 2 2 3 3 4
Q(x) := ______= 1 + l1x - l1x + (2l1 + l2)x (mod x ).
exp(x)
Q
For indeterminates xi of dimension 2 we set := iQ(xi). Denoting by ci the i*
*-th elementary
symmetric function in the xione gets (using the definition of the Hazewinkel ge*
*nerators)
v1
(2)= __c1
2
v21 2
(4)= __(3c2- c1) and
4
v31 3 v2 3
(6)= __(4c1- 13c1c2+ 16c3) + __(c1- 3c1c2+ 3c3).
8 2
18
Beta-elements and divided congruences
2,4
To prove part a) one has that BPQ =(BP Q Q + Q BPQ) is a one-dimensional Q*
*-vector space
generated by v1 v1. Moreover, one checks that the image of BP 4BP is generate*
*d (over Z(2)) by
v1_v1_ v1_v1_ 2
2 and that 4 is a representative of ff1. To see the latter, observe that *
*ff1 is represented by
t1, hence [R , A.1.2.15] ff21is represented in the cobar complex by t1 t1 = (1*
* t1)(t1 1), and then
one applies the description of ffi-1 given in section 2.2. Using the notations *
*introduced in [L2], one
computes that
v1 v1 2,4 (4)
KBP<2>(T X)(4)= c(0)1c(1)1____ inBP Q =(BP Q Q + Q BPQ) ,
4
hence the image of the corner of X in Ext2,4is represented by q_2. v1_v1_2= q .*
* ff21. The final assertion
follows because the only non-trivial element of sss2has Adams-Novikov filtratio*
*n precisely 2.
For part b), we know that Ext2,8is generated by ff1__ff3and fi2,2. As fi2,2is a*
* permanent cycle in
the ANSS whereas ff1__ff3is not we know that the image of X in Ext2,8is a multi*
*ple of fi2,2. One
computes that in the notation of section 2.2 ffi(v22) is represented by z = t41*
*+v21t21in C1(A=2). Hence
fi2,2= ffi0ffi(v22) is represented in BP Q2,8=(BP Q Q + Q BPQ)(8)by -1_8(v1 *
* v31) + 5_16(v21 v21) -
3_ 3 3 2 2 3 4
8(v1 v1). Computing enough of the image of v1t1, v1t1, v1t12and2t1 under BP8BP *
*,! BP8BP Q '
BP Q2,8-! BP Q2,8=(BP Q Q + Q BPQ )(8), one sees that v1_v1_82 im (BP 8BP) *
*and fi2,2is
2 v2
represented by v1__1_16. We provide the following argument for general n, for t*
*he proof here wenneed
the casenn = 3. Observe that by the computations of the previous sections, the *
*image of Ext2,2in
BP Q2,2=(BP Q,2n Q+Q BP Q,2n+BP 2nBP) is given by representatives consisting of*
* summands of
the form vi1vk2 vj1with rational coefficients. Moreover, these elements map und*
*er the isomorphism OE
of section 2.2 to polynomials in v1, v2 and t1. In other words, the f-invariant*
* in bidegree (2, 2n) factors
through the subgroup generated by elements vi1vk2 vj1modulo elements of the fo*
*rm OE-1(vi1vj2tk1),
1 vj1and vi1vj2 1. Denote this quotient by B2n. A computation using the resu*
*lts of the previous
section shows that __c_22n-1v2n-21 v2n-21is not zero in B2n for c an odd integ*
*er. More precisely, no
elementnin the relations defining the quotient B2ncontains such a summand. Amon*
*g the elements in
Ext2,2, the image of fi2n-2,2n-2in B2ncontains such a summand and the other gen*
*erators exhibited
n
in Proposition 15 do not. Thus we have a well-defined map Ext2,2! Z=2 given by *
*mapping an
element to 1 if it admits a representative in B2n having a summand __c_22n-1v2n*
*-21 v2n-21with c odd.
This map is a projection to fi2n-2,2n-2. We would like to consider the summands*
* of
KBP<2>(T X)(8) (2) (6)+ (4) (4)+ (6) (2)mod (BP Q Q + Q BPQ)(8)
as elements in B23. Once this is achieved, we have to consider only the summand*
*s involving v21 v21.
The only summand which is not already given by a representative in B23is c(0)1(*
*c(1)31- 3c(1)1c(1)2+
3c(1)3)v1_v2_4in (2) (6). One computes (for p = 2 and the Hazewinkel generat*
*ors as before) that
3 v1 v2 v2 v1 v3 1 *
* v v v2 v2
OE-1(t2) = 1_v2_2- v2_1_1+ 1_v1_4- ____18+ _1__4-3_1__8. Hence we have OE-1(t1t*
*2) = -_1__24- _1__1_16+...,
2 v2 *
* (0) (1)3
so we have to look at the coefficients of v1_v2_4and v1__1_16which by the above*
* equal c1 (c1 -
3c(1)1c(1)2+ 3c(1)3) and (3c(0)2- c(0)21)(3c(1)2- c(1)21). We also have that c(*
*0)1(c(1)31- 3c(1)1c(1)2+ 3c(1)3)
equals c(0)1(c(1)31+ c(1)1c(1)2+ c(1)3) and (3c(0)2- c(0)21)(3c(1)2- c(1)21) eq*
*uals (c(0)2+ c(0)21)(c(1)2+_c(1)21)
modulo 2 when evaluated on [X]. Now the assertion follows as in part a). *
* |__|
Of course, it is possible to do similar but more complicated computations for 2*
*n-dimensional
(U, fr)2-manifolds in case n > 4. We always have a projection to Z=2 looking at*
* the power of
2 in the denominator of the coefficient of the summand v2n-21 v2n-21. The comp*
*utation then re-
duces to compute those (2i)which contribute to this summand. The diligent read*
*er may thus
19
Jens Hornbostel and Niko Naumann
prove statements of the following form: The element h2n-1survives (equivalently*
*: there is a framed
manifold in dimension 2n - 2 having Kervaire invariant 1) if and only if < Fn(c*
*(0)i, c(1)j), [X] > is
odd for a certain explicit polynomial Fn. The main problem in the computation o*
*f Fn is to find
representatives in B2n (that is sums of vi1vk2 vj1) for elements arising in th*
*e (2n-i) (i). For
n = 3 this was done using t1t2. In the case n > 3, it will be necessary to find*
* suitable elements in
BP 2nBP which will involve tifor larger i and the computation of OE-1 of these *
*elements.
The rest of this section is devoted to the computation of the f-invariant in di*
*mensionn2n+at1the
prime 2. More generally, we compute the f-invariant of fis2n,2n2 Ext2,(3s-1)2fo*
*r all n > 0 and
s > 1 odd.
We use the notations of section 3.1 for p = 2 and N = 3 and those of section 3.*
*2 and write
T := a1-1_22 D=2D which is an Artin-Schreier generator for the extension V1,0= *
*V1,1' k[a3] V1,2.
Theorem 19. The image of the f-invariant in V1,2is given by
f(ff1ff2n+1-1) = T forn > 0,
f(fis) = 1 + as3fors > 3 odd ,
f(fis2,2) = 1 + a2s3fors > 1 odd and
n 3.2n-2s
f(fis2n,2n) = (a23+ a3 ) fors > 1 odd andn > 2.
Proof.For the first line, recall that mod 2 we have fft:= fft,1:= ffi(vt1). One*
* computes that in the
cobar complex ff1fftis represented by t1 1_2[(2t1+ v1)t- vt1] , use the descri*
*ption of the product in
the cobar complex of [R , A.1.2.15]. Using that OE is a ring isomorphism and th*
*e description of ffi-1,
see section 2.2, one further computes that ff1fft is represented by -1_4v1 vt*
*1in the usual quotient
of BPQ2, c.f. (7).
The second line is a special case of Theorem 10 (use Proposition 9) and implies*
* the third line because
x1 x20= v22mod v21, recall the invariant sequences (2, v2n1, xn) from section*
* 2.1.
The only case requiring a longer computation is f(fi4,4):
In the notation of section 2.1 we have r = 1, s = t = 4 and
x2 = v42- v31v322 H0,24(BP =(2, v41)).
This value of x2 follows from the definition in [MRW , p. 476] or [R , Theorem*
* 5.2.13])_after cancelling
all possible multiples of v41. One computes that, in the notation of section 2.*
*2, z 2 ( =2)16is given
as
z = t81+ v41t41+ v22t1(t1+ v1) + v1v2t21(t21+ v21) + v21t31(t1+ *
*v1)3
= v1v22t1+ v22t21+ v31v2t21+ +v51t31+ v1v2t41+ v31t51+ v21t61+ *
*t81.
__
One then computes that f(fi4,4) = (w), where w 2 is a lift of z as in sectio*
*n 2.2, is as claimed,
using the relation in Proposition 9 and that q0(a1) = q0(a3) = 1 mod 2 (see Pro*
*position 7).
Now the value for f(fis4,4) for any s follows immediatelyPfrom the derivation p*
*roperty of the con-
necting homomorphism ffi, namely ffi(xn) = ffi(x)( n-1i=0jR(x)ijL(x)n-1-i). Al*
*ternatively, f(fis4,4)
may be computed directly for any odd s using that (q0 id)jL(x2) = 0 mod 2, jR *
*and OE-1 are ring
homomorphisms and Proposition 9. We obtain f(fis2n,2n) = f(fis4,4)n-2 for all s*
* and n_>_3 because
xn = x2n-1for all n > 3. *
* |__|
Fix some n > 3. To explain the relevance of the above computation for the probl*
*em of projecting
the f-invariant to fi2n-2,2n-2we contemplate the following diagram.
20
Beta-elements and divided congruences
_________ss0________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*__________________________________________@
____________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*______________________________________
___"f_______________________________&&________________*
*_________
(19) Ext2,2n[BPO]__//_iCpD__2n-1OOQ=Z[2] Z=2" `
CCC (17)OO| ||
CCC || |
CCC O_"(18)//_ |
'CCCCCD2n-1OOQ=Z[2] D=2DOO ||
CCC || (section3.||1)||
C!!C?O| " ?O| fflffl|
V"1,2O________//V1,2ss_//k = F4.
By the results in section 3.1, V1,2is k-free on the set {ai3T j|i > 0, j = 0, 1*
*} and ss is defined to be the
projection to the coefficient of a2n-23. The map ss0is defined to be the projec*
*tion to the generator
fi2n-2,2n-2, c.f. Proposition 15. Theorem 19 determines representatives in V"1,*
*2:= D2n-1 Q=Z[2]\V1,2
n
for all generators of Ext2,2[BP ] and thus defines the map '. We know that (19)*
* commutes when ss
and ss0are omitted.
Theorem 20. The diagram (19) is commutative.
*
* n
Proof.By inspection of Proposition 15 and Theorem 19 , the only generator of Ex*
*t2,2[BP ] whose__
f-invariant contains a2n-23is the Kervaire element fi2n-2,2n-2. *
* |__|
Corollary 21. Let n > 3 and X a (U, fr)2-manifold of dimension 2n. Then the cor*
*ner of X
has Kervaire invariant one if and only if the f-invariant of X admits a represe*
*ntative in V"1,2which
contains the summand a2n-23.
Note that the coefficient of a2n-23in the f-invariant of X can rather easily be*
* expressed in terms of
Chern numbers of X. The very reason that this does not give us the Chern number*
*s determining
the Kervaire elements is the indeterminacy in the above constructions caused by*
* the projection
D2n-1 Q=Z[2] -! D__2n-1 Q=Z[2].
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Jens Hornbostel and Niko Naumann jens.hornbostel@mathematik.uni-regensburg.de
niko.naumann@mathematik.uni-regensburg.de
NWF I- Mathematik, Universit"at Regensburg, 93040 Regensburg
22