A modular description of the K(2)-local sphere
at the prime 3
Mark Behrens 1
Department of Mathematics, MIT, Cambridge, MA 02139, USA
_____________________________________________________________________________
Abstract
Using degree N isogenies of elliptic curves, we produce a spectrum Q(N). This
spectrum is built out of spectra related to tmf. At p = 3 we show that the K(2)-
local sphere is built out of Q(2) and its K(2)-local Spanier-Whitehead dual. Th*
*is
gives a conceptual reinterpretation a resolution of Goerss, Henn, Mahowald, and
Rezk.
AMS classification: Primary 55Q40, 55Q51, 55N34. Secondary 55S05, 14H52.
Key words: Chromatic filtration, topological modular forms, cohomology
operations.
_____________________________________________________________________________
Contents
Part 1: The spectrum Q(N) 7
1.1 A simplicial stack 8
1.2 Realizing the simplicial stack 11
1.3 0(2) modular forms 14
1.4 The Adams-Novikov E2-term of Q(2) 17
1.5 Computation of the maps 20
1.6 Extended automorphism groups 23
1.7 Effect on Morava modules 28
1.8 Calculation of V (1)*(Q(2)) at p = 3 31
Part 2: The 3-primary K(2)-local sphere 35
_______
1 The author is partially supported by the NSF.
Preprint submitted to Elsevier Preprint 6 June 2005
2.1 Duality pairings 36
2.2 K(2)-local equivalences 39
2.3 LK(n)S is hyperbolic 39
2.4 Identification of IK(2)T MF 47
2.5 Identification of I-1K(2)^ T MF 51
2.6 Proof of Theorem 2.0.2 52
__
2.7 A spectrum which is half of S 53
2.8 Recollections from [16] 55
__
2.9 A Lagrangian decomposition of S 57
__
2.10 Building Q from Q 58
2.11 Proof of Theorem 2.0.1 62
__
Appendix A: A cellular model for S 66
References 69
Introduction
Goerss, Henn, Mahowald, and Rezk [16] have produced at the prime 3 a tower
of spectra of the form
T MF ! T MF _ 8E(2) ! 8E(2) _ 40E(2)
! 40E(2) _ 48T MF ! 48T MF.
Here, as is the case everywhere in this paper unless indicated otherwise, we are
working in the K(2)-local category, and everything has been implicitly K(2)-
localized. The authors of [16] show that the tower refines to a resolution of
the K(2)-local sphere at the prime 3. This means that there exists a diagram
LK(2)S oo______X1 oo________X2 oo________X3 oo_________X4
| | | | ||
| | | | ||
| fflffl| fflffl| fflffl| ||
fflffl| -1TMF_ 6E(2)_ 37E(2)_ ||
T MF 7E(2) 38E(2) 45TMF 44T MF
such that each spectrum Xi is the fiber of the ith vertical map, and such that
the connecting morphisms of these fiber sequences give the maps in the tower.
2
This resolution helps organize the computation of Shimomura and Wang of
ss*(LK(2)S0). We shall refer to this as the GHMR resolution.
The spectrum tmf is the connective spectrum of topological modular forms.
We write T MF for the non-connective spectrum tmf[ -1], but this distinc-
tion is irrelevant, since we have
LK(2)tmf ' LK(2)T MF ' EO2 = EhG242.
Here E2 is Morava E-theory and G24 is a maximal finite subgroup of the
extended Morava stabilizer group G2 at the prime 3. The spectrum EO2 exists
by the Hopkins-Miller theorem [40], and is E1 by the machinery of Goerss
and Hopkins [17]. The equivalence LK(2)T MF ' EO2 is discussed in greater
detail in Remark 1.7.3.
Remark. It has become standard to let E2 denote the Landweber exact co-
homology theory corresponding to the Lubin-Tate deformation of the Honda
height 2 formal group F2, defined over Fp2, with p-series
2
[p]F2 = xp .
However, for the 3-primary applications in this paper, we shall let E2 denote
the spectrum corresponding to the Lubin-Tate deformation of the height 2
formal group F20over F9 whose 3-series is given by
[3]F02= -x9.
Our reason for doing this is that F20is the formal group of the unique super-
singular elliptic curve over F9 (see Section 1.6, in particular Remark 1.6.4).
The GHMR resolution is a generalization to chromatic level 2 of the J-
spectrum fiber sequence
_N -1
LK(1)S0 ! KOp --- ! KOp.
That this gives the K(1)-local sphere was first observed in unpublished work of
Adams and Baird, and an independent verification of was provided by Ravenel
[38], [6]. Here N is a topological generator of Zxp.
There is a spectrum T MF0(2) which is an analog of T MF for the congruence
subgroup 0(2) < SL2(Z). We have equivalences
LK(2)T MF0(2) ' EhD82' LK(2)(E(2) _ 8E(2))
3
where D8 < G2 is a dihedral group of order 8. After inverting 6, the spectrum
T MF0(2) coincides with the elliptic cohomology theory Ell of Landweber,
Ravenel, and Stong [28].
The GHMR tower may be made less efficient, to take the form
T MF ! T MF _ T MF0(2) ! T MF0(2) _ 48T MF0(2)
! 48T MF0(2) _ 48T MF ! 48T MF
by letting the extra copies of E(2) kill themselves pairwise. It is this resolu*
*tion
that we shall endeavor to reinterpret.
The GHMR resolution was produced using the work of Devinatz and Hop-
kins [12], where the K(2)-local sphere is identified as a homotopy fixed point
spectrum
EhG22' LK(2)S0.
The authors of [16] produced a resolution of the trivial G2-module by per-
mutation modules. The authors then realize their resolution to produce the
tower.
Mahowald and Rezk wanted a modular description of the GHMR resolution.
The motivation is that the Adams operation _N corresponds to the Nth power
isogeny on the multiplicative group Gm . By replacing the multiplicative group
with an elliptic curve, one can instead consider certain degree N isogenies of
elliptic curves. Mahowald and Rezk studied the corresponding map
"_N -1"
T MF --- - - !T MF0(N)
at the primes 2 and 3 [30], [29]. This setup is complicated by the fact that th*
*ere
is a whole moduli space of elliptic curves, and elliptic curves do not support
unique degree N isogenies. The map that Mahowald and Rezk studied is a
projection of the first map of the GHMR resolution. Ando, Hopkins, and
Strickland have also extensively studied operations on elliptic cohomology
theories arising from isogenies of elliptic curves [1], [2], [3].
The aim of this paper is expand on the work of Mahowald and Rezk to produce
a tower
T MF ! T MF0(N) _ T MF ! T MF0(N)
which refines to a resolution of an E1 ring spectrum Q(N). Here N is prime
to p. This tower arises conceptually from certain degree N isogenies of elliptic
curves. Rezk had considered a similar setup from the point of view of flags
of subgroups of an elliptic curve. For a spectrum X, let DK(2)(X) denote the
Spanier-Whitehead dual in the K(2)-local category
DK(2)(X) = F (X, LK(2)S).
4
We conjecture that for appropriate N, the spectrum Q(N) is half of the K(2)-
local sphere.
Conjecture 1 Let p be greater than 2. If N is chosen to be a topological
generator of Zxp, then the natural sequence
Dj j
DK(2)Q(N) -! LK(2)S -! Q(N)
is a cofiber sequence. Here j is the unit of the K(2)-local ring spectrum Q(N).
If p = 2, and N is a topological generator of an index 2 subgroup of Zx2, then
there is an index 2 subgroup eG2 of G2 and a cofiber sequence
Dj heG2 j
DK(2)Q(N) -! E2 -! Q(N).
Note that the group Zx2is not topologically cyclic, but it does have topologi-
cally cyclic index 2 subgroups.
The main result of this paper is to verify Conjecture 1 at the prime 3, with
N = 2 (Theorem 2.0.1). The author also knows Conjecture 1 to hold in the
case p = 5 and N = 2. The author has no evidence for the invariance of
Conjecture 1 under different choices of N.
This approach to understanding the K(2)-local sphere has several advantages.
The maps building Q(N) are quite computable using well known formulas for
Weierstrass curves and have very natural descriptions in terms of isogenies of
elliptic curves. The cofiber sequence of Conjecture 1 explains the self-duality
of the GHMR resolution, and an identification of DK(2)T MF explains the ap-
pearance of the suspension 48. The very difficult computations of Shimomura
and Wang of the homotopy of LK(2)S should be verified independently using
our decomposition. As a side-effect of our work we also reproduce the short
tower (Proposition 2.9.1)
T MF ! 8E(2) ! 40E(2) ! 48T MF (0.0.1)
__ 1
which refines to the spectrum S = EhG2 given in [16]. Here G12is the kernel of
the reduced norm
G2 ! Z3.
We are not able to describe the connecting map
Q(N) ! DK(2)Q(N)
of Conjecture 1, nor are we able to describe the middle map of the short tower
(0.0.1).
5
Our approach to proving Theorem 2.0.1 is computational, and this is the
reason for our specialization to the prime 3 and N = 2. It would be nice to
have a conceptual and elegant proof of Conjecture 1 for all primes and all
N. The 2-primary applications with N = 3 should be rewarding. There is
essentially nothing to be gained computationally from Conjecture 1 if p 5.
This paper is organized into two parts. A detailed outline of the content is
given at the beginning of each part. In Part 1 we give a construction of Q(N),
and quickly specialize to the case N = 2 and p = 3. In this case we give
a computation of the V (1)-homology groups V (1)*Q(2) where V (1) is the
Smith-Toda complex. In Part 2 we prove Conjecture 1 in the case N = 2 and
p = 3 (Theorem 2.0.1). The proof uses the V (1) homology computations given
in Part 1 as well as the computation of the V (1) homology of LK(2)S0 _ and
thus does not generalize to arbitrary N and p.
Everything in this paper is implicitly K(2)-localized unless specifically speci*
*fied
otherwise. Sections for which this convention does not hold are indicated as
such in their beginnings.
We highlight a few aspects of this paper which may be of independent in-
terest. The duality decomposition of Conjecture 1 may be interpreted as a
Lagrangian decomposition for a hyperbolic duality pairing on the K(2)-local
sphere LK(2)S. The abstract framework of duality pairings and Lagrangians
in a triangulated symmetric monoidal category is given in Section 2.1.
In Section 2.3, we show that for arbitrary n, at_an arbitrary prime p, the
1
canonical pairing on LK(n)S is hyperbolic, and S =_EhGnnis_a Lagrangian. We
identify the K(n)-local Spanier-Whitehead dual of S by proving there is an
equivalence __ __
DK(n)(S ) ' -1S .
__
Finally, for computational reasons, we need a cellular decomposition of S in
the K(n)-local category. This task is relegated to Appendix A, where we prove
that there is a (K(n)-local) cellular decomposition
__ 0 0 0
S ' S [i e [i e [i . ...
The author would like to thank Charles Rezk and Mark Mahowald, who shared
their ideas and preliminary notes so generously. Thanks also go to Mike Hop-
kins for suggesting the duality mechanism which is the main result of this
paper, Paul Goerss for explaining En-homology operations, Sharon Hollan-
der and Tilman Bauer, for explaining various aspects of stacks, and Nasko
Karamanov for sharing his computational knowledge. Daniel Davis provided
valuable assistance with homotopy fixed point spectra, in particular with The-
orem 2.3.2 and Lemma A.0.4. Many ideas in this paper are culled from vari-
6
ous works of Matthew Ando and Neil Strickland. Discussions with Peter May,
Haynes Miller, and Doug Ravenel during the course of this project were very
helpful. The initial version of this paper dealt with the Galois action on the
Morava stabilizer group improperly. The author was alerted to this problem
by Hans-Werner Henn, and Mike Hopkins' help in sorting it out was invalu-
able. Thanks also go to the referee for pointing out that the hypotheses of
Proposition 2.4.4 needed modification.
Part 1: The spectrum Q(N)
This part is organized as follows. In Section 1.1 we produce a simplicial stack
based on certain degree N isogenies.
In Section 1.2 we topologically realize our simplicial stack to a cosimplicial
spectrum whose totalization is our spectrum Q(N). We give two approaches
to this realization problem. The first is based on a sheaf of E1 ring spectra
produced by Hopkins and his collaborators. The second, specialized to p = 3
and N = 2, uses the Goerss-Hopkins-Miller theory and a supersingular elliptic
curve.
In Section 1.3 we give a description of the ring of 0(2) modular forms in
terms of Weierstrass equations. The results of this section are probably well
known.
Our computations of the homotopy of Q(2) are based on the ANSS. In Sec-
tion 1.4, we describe a chain complex which computes the Adams-Novikov E2
term for Q(2). This ANSS E2-term is the hypercohomology of the simplicial
stack we constructed in Section 1.1.
Section 1.5 is devoted to explicitly computing the maps on rings of modular
forms induced by the face maps of our simplicial stack. From these maps we get
the differentials in the chain complex of Section 1.4. These explicit formulas
are also necessary to complete the construction of Q(2) given in Section 1.2.1.
In Section 1.6 we describe explicitly the automorphism groups of a supersin-
gular elliptic curve C and its formal group C^ . In Section 1.7 we use this
description to describe the effects that our maps have on E2-homology. These
formulas will be used in many technical lemmas in Part 2, and also may be
used in the construction of Q(2) given in Section 1.2.2.
Our approach to Theorem 2.0.1 is computational and based on V (1)-homology
computations. In Section 1.8, we compute the V (1)-homology of Q(2). This
computation is compared to computations of Goerss, Henn, Mahowald, and
7
Shimomura.
1.1 A simplicial stack
Let M be the moduli stack of non-singular elliptic curves over Z(p). Let ! be
the line bundle of invariant differentials over M, so that the sections of ! k
give weight k weakly modular forms.
H0(M, ! k ) = MFk
MF* is the ring of weakly modular forms over Z(p)[8].
MF* = Z(p)[c4, c6, 1]=(c34- c26= 1728 )
We must invert because we are taking sections over the complement of the
singular locus of the moduli space of generalized elliptic curves. We shall ref*
*er
to the ring of modular forms (without inverting the discriminant) as mf*.
Given an elliptic curve C, we shall mean by a 0(N) structure a chosen discrete
subgroup of C which is isomorphic to CN , the cyclic group of order N, but we
do not fix the isomorphism. 0(N) structures are in one to one correspondence
with degree N isogenies with cyclic kernels. Given a 0(N) structure H of C,
one has an isogeny
OEH : C ! C=H
and given such an isogeny, one recovers the 0(N) structure by taking the
kernel. We define M0(N) to be the moduli stack of non-singular elliptic curves
over Z(p)with 0(N) structure. We shall always be considering N prime to p.
Moduli stacks of elliptic curves with additional structure have been widely
studied by arithmetic algebraic geometers. See, for example, [9], [27].
Given an integer N, every elliptic curve has an Nth power endomorphism
[N] : C ! C
given (for positive N) by
[N](P ) = P_+_._.+.P_-z_____".
N
It is an isogeny of degree N2.
Given any degree N isogeny OE, there exists a dual isogeny bOE. The dual isogeny
has the property that
ObEO OE = [N].
8
See, for instance, [45, III.6].
We shall now describe a simplicial stack Mo . Actually Mo is a semi-simplicial
stack since we do not use degeneracies. It has only 0, 1, and 2 simplices. We
will describe this simplicial object first in conceptual terms, and then in pre*
*cise
stack theoretic language. The simplices will be given as follows:
o The 0-simplices are elliptic curves.
o The 1-simplices are certain isogenies. There will be two types:
. Degree N isogenies OEH with cyclic kernel H. These are in one-to-one cor-
respondence with elliptic curves C with a 0(N) structure H.
. The endomorphisms [N]. These are in one-to-one correspondence with
elliptic curves, since every elliptic curve possesses a unique such endomor-
phism.
o The 2-simplices correspond to relations of the form bOEHO OEH = [N]. These
relations are in one-to-one correspondence with elliptic curves with a 0(N)
structure H, since there is one such relation for every isogeny OEH .
The semi-simplicial stack Mo is a diagram
oo_d0__ ` ood0____
M oo_d1__M M0(N) ood1___M0(N)_d
oo_ ___ 2
in the category of stacks. Given a Z(p)-algebra R, the R-points of M is a
groupoid of elliptic curves C over R, and the R-points of M0(N) is a groupoid
of pairs (C, H) of elliptic curves C over R with 0(N) structure H C(R).
The simplicial stack Mo should be viewed on the level of R-points as consisting
of moduli of diagrams of the following form.
8 9
ae oe >>> >>>
OEH d0 >>>> >>>>
-d0 (C, H) -! C=H - >>>< >>>=
[N]
{C} d1 ` d1- > (C, H)C__________//_CBB>
- ae [N] oe d2 >>> CCCC >>>
C --! C - >>>> OEHCCC bOE >>>
>>: !! H>>>
C=H ;
The face maps are what one might expect from taking the nerve of a category.
To give them we must define some maps of stacks. These maps of stacks are
given on the level of R-points by the formulas below.
o Forget 0(N) structure:
OEf : M0(N) ! M
(C, H) 7! C
9
o Quotient out by level 0(N) structure:
OEq : M0(N) ! M
(C, H) 7! C=H
o Codomain of the Nth power endomorphism:
_[N]: M ! M
C 7! C=C[N]
o Dual 0(N) structure (the dual 0(N) structure cH is defined such that
OEHb= bOEH):
_d : M(N) ! M(N)
(C, H) 7! (C=H, cH)
We comment that there is a canonical isomorphism
rC : C=C[N] ~=C.
This isomorphism is not an equality, and this distinction is important in the
context of stacks. We may use this isomorphism to define _[N]on M0(N) by
_[N]: M0(N) ! M0(N)
(C, H) 7! (C=C[N], r-1C(H)).
We note that we have the following relations
OEq(C) = OEf(_d(C)) (1.1.1)
_d O _d = _[N]. (1.1.2)
The first relation indicates that if we have defined OEf and _d, then OEq is
automatically determined.
The careful reader will be bothered that the maps OEf, _[N], and _d are not
defined above in a precise manner. In Section 1.5, we give explicit formulas
for N = 2 and p = 3 for these maps on the level of Hopf algebroids. The
stackifications of these Hopf algebroids are M and M0(2), so we get induced
maps of stacks. We can then check relations 1.1.1 and 1.1.2 explicitly. Since
we are only concerned with the case p = 3, N = 2 in this paper, we choose to
not elaborate further on the general case.
We use these`maps of stacks to give the face maps of Mo. The face maps
di : M M0(N) ! M are defined by
a
d0 = _[N] OEq
a
d1 = IdM OEf
10
`
The face maps di : M0(N) ! M M0(N) are defined by
d0 = _d
d1 = OEf
d2 = IdM0(N)
The simplicial identities are verified by relations 1.1.1 and 1.1.2.
1.2 Realizing the simplicial stack
We wish to topologically realize the simplicial stack Mo as a cosimplicial
E1 ring spectrum Q(N)o. Actually, Q(N)o is a semi-cosimplicial E1 ring
spectrum, since we have no codegeneracies.
___d1__//_ ___d___//2_//_
T MF ___d0__//_T MF x T MF0(N) dd1 T MF0(N)
___0___//
We then define the spectrum Q(N) by
Q(N) = T ot(Q(N)o).
The spectrum Q(N) is an E1 ring spectrum since the coface maps are maps
of E1 ring spectra.
We give two approaches to constructing Q(N)o. The first is based on a con-
struction of tmf due to Hopkins and his collaborators as the global sections of
a sheaf of E1 ring spectra. Since this work is unpublished, we give an alterna-
tive construction for p = 3 and N = 2 using the Goerss-Hopkins refinement of
the Hopkins-Miller theorem. The latter approach is sufficient for our purposes
since we are working K(2)-locally.
1.2.1 Sheaf theoretic construction
Hopkins and his collaborators have constructed a sheaf Oellof E1 ring spectra
on M in the 'etale topology. Paul Goerss has written a useful survey of this
point of view [14]. The sheaf Oellhas the property that if C is any elliptic
curve over R which is 'etale over M, then the homotopy groups of the sections
over C are given by
ss2k(Oell(C)) ~=!Ck
where !C is the free R-module of holomorphic 1-forms on C. One recovers the
spectra T MF and T MF0(N) as the sections of this sheaf over the appropriate
11
moduli stacks.
T MF = Oell(M)
T MF0(N) = Oell(M0(N))
Since Oellis a sheaf in the 'etale topology, we need the map
OEf : M0(N) ! M
to be 'etale to take sections over M0(N). This is why we insist that p does not
divide N.
Our requirement that p does not divide N also implies that the maps
_[N]: M0(N) ! M0(N)
_[N]: M ! M
are isomorphisms. Relation 1.1.2 implies that
_d : M0(N) ! M0(N)
is an isomorphism. We have already indicated that the map OEf is 'etale. Rela-
tion 1.1.1 indicates that OEq is 'etale. We therefore get induced maps
OE*f:T MF = Oell(M) ! Oell(M0(N)) = T MF0(N)
OE*q:T MF = Oell(M) ! Oell(M0(N)) = T MF0(N)
_*[N]:T MF = Oell(M) ! Oell(M) = T MF
_*d:T MF0(N) = Oell(M0(N)) ! Oell(M0(N)) = T MF0(N).
These maps induce the coface maps di of Q(N)o.
1.2.2 Supersingular construction
We now give an alternative construction of Q(2)o at p = 3 that uses computa-
tions of endomorphisms of the supersingular elliptic curve of Section 1.6. This
approach to T MF mirrors the approach in [21]. The Goerss-Hopkins-Miller
theorem [40], [17] states that there is a contravariant functor
E : FGL n ! E1 ring spectra
where FGL n is the category of pairs (k, F ) where k is a perfect field of cha*
*r-
acteristic p and F is a formal group law of height n over k. Consider the
supersingular elliptic curve
C : y2 = x3 - x
12
over F9. The Weierstrass curve C has a canonical coordinate, and gives rise
to a formal group law C^ of height 2. The curve C has a 0(2) structure H
generated by the point (x, y) = (0, 0) (see Section 1.3). In Section 1.7, we
produce an explicit degree 2 isogeny
_H : C ! C
with ker _H = H. We have extended automorphism groups (including au-
tomorphisms of the ground field F9) of elliptic curves (respectively, elliptic
curves with 0(2) structures) given by
Aut =F3(C) = G24
Aut =F3(C, H) = D8.
Here G24 and D8 correspond to certain subgroups of G2. These subgroups are
described explicitly in Sections 1.6 and 1.7. The map _H is invariant under
D8, since every element of D8 fixes the 0(2) structure H.
The value of the Goerss-Hopkins-Miller functor E on (F9, C^ ) is the version
of Morava E-theory that we will be using.
E(F9,C^)= E2
The map _H induces an isomorphism of formal group laws
b_H : C^ ! C^
invariant under the action of D8, and the Goerss-Hopkins-Miller theorem im-
plies that it induces a map of E1 ring spectra
_*d: T MF0(2) = EhD82! EhD82= T MF0(2).
We verify directly in Section 1.7 that (_*d)2 = _*[2], where _*[2]is the Adams
operation corresponding to the action of the element 2 2 WxF9 S2 of the
Morava stabilizer group. Define OE*fto be the restriction
OE*f= ResG24D8: T MF = EhG242! EhD82.
We get OE*qfor free by defining it to be the composite
OE*f _*d
OE*q: T MF -! T MF0(2) -! T MF0(2).
We now can define the coface maps diin terms of OE*f, _*[2], _*d, and OE*qas be*
*fore,
and we get our cosimplicial spectrum Q(2)o.
The idea is that K(2)-locally, we only need to consider sections of Oellin a
formal neighborhood of the supersingular locus, and that is precisely what the
Goerss-Hopkins-Miller theorem is implicitly doing at chromatic level 2.
13
1.3 0(2) modular forms
1.3.1 Moduli description of mf0(2)
In order to compute the maps in the resolution for p = 3 and N = 2 we need
to know something about 0(2) modular forms. Most of the computations in
this section have been carried out by Mark Mahowald and Charles Rezk, or
are just in the literature on modular forms. It is well known (see for instance,
the appendix of [18]) that the ring of 0(2) modular forms is given (with 2
inverted) by
mf0(2)*[1=2] = Z[1=2][ffi, ffl]
where the weights are given by
|ffi|= 2
|ffl|= 4.
The discriminant is given by
= -64ffl2(ffl + ffi2).
Denote the ring with inverted by
MF0(2)*[1=2] = mf0(2)*[1=2, -1].
Let M0(2) be the moduli stack of non-singular elliptic curves with 0(2) struc-
ture. The ring MF0(2) should be interpreted as sections of tensor powers of
the line bundle ! over this stack.
MF0(2)k = H0(M0(2), ! k )
We can eliminate our use of the line bundle ! if we add the data of a non-zero
tangent vector to the structures on elliptic curves we are taking moduli of.
Namely, let M10(2) be the moduli stack of non-singular elliptic curves with
the data of a 0(2) structure and a non-zero tangent vector at the identity.
Then we have
MF0(2)* = H0(M10(2), O)
where O is the structure sheaf of M10(2). There is an action of the multiplica-
tive group Gm on M10(2); the action is given by the multiplication action on
the chosen tangent vector. Under this action, the sections above break up into
a direct sum as the weights of the Gm representations. These coincide with
the weights of the modular forms.
14
1.3.2 Calculation of MF0(2)*[1=2] using Weierstrass curves
We shall give an alternate description of the generators of MF0(2)*[1=2] using
Weierstrass equations. For the rest of this section we will be implicitly worki*
*ng
with 2 inverted. Under this condition, every Weierstrass curve is isomorphic
to one of the form [45]
Cb : y2 = 4x3 + b2x2 + 2b4x + b6.
This curve is nonsingular if the discriminant
1 3 3 1 2 2
= -27b26+ (9b2b4 - __b2)b6 - 8b4 + __b2b4
4 4
is a unit. We shall only consider non-singular curves. We implicitly think
of these curves as coming with the data of the tangent vector @=@z where
z = x=y. The only transformations which leave this form of equation and
tangent vector invariant are those of the form
Or : x 7! x + r, y 7! y.
A 0(2) structure is the choice of a point of exact order 2 in Cb. These points
coincide with the points (x, y) of the curve Cb with y = 0. Thus a 0(2)
structure consists of a chosen root e1 of the cubic 4x3 + b2x2 + 2b4x + b6. The
curves Cb with this additional data may be written in the form
Cfl,e1: y2 = 4(x - e1)(x2 + fl2x + fl4)
where we have
b2= 4(fl2 - e1)
b4= 2(fl4 - e1fl2)
b6= -4fl4e1.
Given the root e1 and the coefficients b2, b4, b6, the coefficients fl2 and fl4*
* may
be expressed as
b2 + e1
fl2= _______
4
4b4 + e1b2 + e21
fl4= _______________.
8
The data of the 0(2) structure allows us to remove the automorphism Or.
We simply shift the root e1 to be 0 by the transformation Oe1. This puts Cfl,e1
into the canonical form
Cq : y2 = 4x(x2 + q2x + q4).
15
The curve Cq is regarded as implicitly carrying the data of the tangent vector
@=@z and the 0(2) structure generated by the point (x, y) = (0, 0). After the
transformation Oe1, one has
q2= 2e1 + fl2 (1.3.1)
q4= e21+ fl2e1 + fl4. (1.3.2)
The discriminant of the curve Cq is given by
= q24(16q22- 64q4).
There are no non-trivial automorphisms of the curves Cq preserving the 0(2)
structure and the tangent vector, thus we have shown that the stack M10(2)
is an affine scheme.
M10(2) = spec(Z[1=2, q2, q4, -1])
The ring MF0(2)*[1=2] is just the ring of functions.
MF0(2)*[1=2] = Z[1=2, q2, q4, -1]
1.3.3 Relation to ffi and ffl
We now relate our generators q2 and q4 to the classical generators ffi and ffl *
*that
appear in the Jacobi quartic. Assume we are working over an algebraically
closed field k of characteristic not equal to 2 or 3. An elliptic curve can be
expressed by a Weierstrass equation
Ce : y2 = 4(x - e1)(x - e2)(x - e3)
with e1 + e2 + e3 = 0. (This last condition is equivalent to the Weierstrass
equation taking the form y2 = 4x3 - g2x - g3.) We can take e1 to be the
specified 0(2) structure. In [18], the quantities ffi and ffl are then given by
3
ffi= -__e1
2
ffl= (e1 - e2)(e1 - e3).
By multiplying out the factors containing e2 and e3, we see that Ce, with 0(2)
structure e1, is given by the curve Cfl,e1with
fl2= -e2 - e3
fl4= e2e3.
Using Equations 1.3.1 and 1.3.2, and the relation e1 + e2 + e3 = 0, we see that
q2= -2ffi
q4= ffl.
16
Thus our generators q2 and q4 of mf0(2)[1=2] are, up to scaler multiple, iden-
tical to the classical generators ffi and ffl.
1.4 The Adams-Novikov E2-term of Q(2)
There are several approaches to computing the homotopy groups of the spec-
trum Q(2). One could compute the maps that OEq, OEf, _d, and _[2]induce on
the homotopy groups of T MF and T MF0(2), and then use the Bousfield-
Kan spectral sequence for T ot(Q(2)o), but it is actually easier to compute the
E2-term of the Adams-Novikov spectral sequence (ANSS) for Q(2) and then
add in the Adams-Novikov differentials using the differentials in the ANSS for
T MF . The discussion of the ANSS for T MF in this section follows [41] and
[21]. Everything in this section is implicitly 3-local, though most of what we
say holds if we simply invert 2.
1.4.1 The ANSS E2 term for T MF .
We explain how to compute the ANSS E2-term for T MF using an 'etale cover
of M1. This material has appeared elsewhere, see [21], [41], and [4]. As men-
tioned in Section 1.3.2, over a Z(3)-algebra every elliptic curve is isomorphic
to one of the form
Cb : y2 = 4x3 + b2x2 + 2b4x + b6.
The automorphisms of this elliptic curve which preserve the tangent vector
are of the form
Or : x 7! x + r.
Consider the elliptic curve Hopf algebroid
B = Z(3)[b2, b4, b6, -1]
B = B[r]
which represents the groupoid of such elliptic curves and isomorphisms. The
right unit is determined by how the coefficients of the Weierstrass equation for
Cb transform. Since every isomorphism class of elliptic curve is represented,
the stackification (in the flat topology) of the pre-stack associated to the Ho*
*pf
algebroid (B, B ) is M1.
The Hopf algebroid (B, B ) suffers from the drawback that the natural map
spec(B) ! M1
17
is not 'etale. We instead consider the curves
Cb : y2 = 4x(x2 + q2x + q4).
The automorphisms of these curves which preserve the tangent vector are
those of the form x 7! x + r where r has the property that
r3 + q2r2 + q4r = 0.
__
This groupoid of elliptic curves is represented by the Hopf algebroid (B , __B)
given by
__ -1
B = Z(3)[q2, q4, ]
__ 3 2
__B= B [r]=(r + q2r + q4r).
Proposition 1.4.1 The map
__ 1
fq : spec(B ) ! M
which classifies the curve Cq is an 'etale cover.
Proof. We first point out that the natural map of stackifications
__
stack(B , __B) ! stack(B, B )
is an equivalence of stacks. This follows since given a curve C of the form Cb
over a Z(p)-algebra R, there is a finite faithfully flat extension R0= R[t]=(4t*
*3+
b2t2 + 2b4t + b6) of R so that over R0, C is isomorphic to a curve of the form
Cq under the transformation x 7! x + t.
Therefore we simply must check that the natural map
__ __
spec(B ) ! stack(B , __B)
is an 'etale cover. It suffices to check that the pullback
__
spec( __B)______//spec(B )
| |
| |
fflffl|_ fflffl|_
spec (B )_____//_stack(B , __B)
is an 'etale cover, that is, that __Bis an 'etale extension. This follows from*
* the
fact that since is invertible, q4 is invertible, and therefore_the derivative*
* of
f(r) = r3 + q2r2 + q4r is nonzero modulo all maximal ideals in B .
In particular, fq is a flat cover. Thus we have the following corollary.
18
Corollary 1.4.2 The ANSS E2-term for T MF may be computed as
__ __ * *
H*(M1, O) ~=Ext __B(B , B) ~=H (C ( __B))
__
where C*( __B) is the cobar complex of the Hopf algebroid (B , __B).
Since the stack M10(2) is an affine scheme after inverting 2, the ANSS for
T MF0(2) is concentrated in the zero line. The ANSS collapses to give
ss2k(T MF0(2)) = MF (2)k.
1.4.2 The ANSS E2-term for Q(2)
The ANSS for Q(2) may be obtained by totalizing a cosimplicial Adams-
Novikov resolution for Q(2). Therefore, its E2-term is the hypercohomology of
the simplicial stack Mo. For convenience we choose to work with the simplicial
stack M1owhere all of the instances of M and M0(2) are replaced by M1 and
M10(2), respectively. The resulting spectral sequence takes the form
Hs,t(M1o, O) ) ss2t-s(Q(2)).
The E2 term may be computed via a hypercohomology spectral sequence
H*cosimp(H*(M1o, O)) ) H*(M1o, O). (1.4.1)
Here H*cosimpis the cohomology of the resulting cosimplicial abelian group.
The hypercohomology group H*(M1o, O) is the cohomology of the totalization
of a double cochain complex C*,*(Q(2)) whose horizontal differentials are given
by
__* d0-d1+d2
C*( __B)_d0-d1_//_MF0(2) C ( __B)______//_MF0(2)___________//0_________//_
||
|| || || ||
C0,*(Q(2)) C1,*(Q(2)) C2,*(Q(2)) C3,*(Q(2))
and whose vertical differentials are given by the differentials of the cobar co*
*m-
plex. Here MF0(2) should be_regarded as a cochain complex concentrated in
*
degree zero. The complex C ( __B) is the cobar complex where the differentials
have been given the opposite sign. The coface maps di are lifts of the maps
induced by the simplicial face maps of M1o. In Section 1.5 we shall compute
these maps explicitly.
We shall let the cochain complex C*(Q(2)) be the totalization of the double
complex C*,*(Q(2)). Then we have
H*(M1o, O) = H*(C*(Q(2)))
19
and the hypercohomology spectral sequence is simply the spectral sequence of
the double complex.
1.5 Computation of the maps
In this section we compute the effects of the maps OEq, OEf, _d, and _[2]on the
appropriate rings of modular forms. Actually, since M1 is not a scheme, we will
lift_the maps involving M1 to the prestack associated to the Hopf algebroid
(B , __B). Our computations of these maps will show, as a side-effect, that OE*
*f is
'etale, and _[2]is an isomorphism. This was what was required to topologically
realize the maps on the spectra of sections of Oellin Section 1.2.1. We shall
also see that we have the relation
_2d= _[2].
This appeared as relation 1.1.2, and it is required for the simplicial identiti*
*es
to hold in the semi-simplicial stack Mo.
1.5.1 The map OEf
__
The stack M10(2) is the affine scheme spec(B ). There are no nontrivial auto-
morphisms. Forgetting the 0(2) structure generated by (x, y) = (0, 0) is the
same as allowing the automorphisms of the curve Cq which do not preserve
this level 0(2) structure. Thus the map
OEf : M10(2) ! M1
is induced by the map of Hopf algebroids
__ __ __
(B , __B) ! (B , B)
__
which is the identity on B and maps r to zero. Proposition 1.4.1 implies that
the map OEf is 'etale.
1.5.2 The map _[2]
The map _[2]: M1 ! M1 takes an elliptic curve C and returns the quotient
C=C[2] where C[2] is the subgroup of all points of order 2 of C. The quotient
map C ! C=C[2] is equivalent to the second power map [2] on the elliptic
curve. The second power map on the curve Cq could be regarded as an endo-
morphism of the curve, but the tangent vector @=@z changes, where z = x=y.
Therefore we shall instead regard the map [2] as being a map from Cq to Cq0
20
where the tuple q0 is given by
q02= 22q2
q04= 24q4.
We also must adjust the automorphism x 7! x + r to x07! x0+ r0 where
r0= 22r.
Therefore, the map on Hopf algebroids
__ __
_*[2]: (B , __B) ! (B , __B)
is given by
_*[2](q2)= 22q2
_*[2](q4)= 24q4
_*[2](r)= 22r.
The map
_*[2]: M10(2) ! M10(2)
__
is the map above restricted to B . Each of these instances of _[2]is an isomor-
phism since we can explicitly define the inverse _[1=2]on the Hopf algebroids,
since we are working 3-locally.
1.5.3 The map _d
The dual isogeny map _d : M10(2) ! M10(2) takes a curve with 0(2) structure
(C, H), identifies H as the kernel of a degree 2 isogeny
OEH : C ! C=H,
and returns the elliptic curve with 0(2) structure (C=H, cH) corresponding to
the dual isogeny
ObEH: C=H ! C.
In our case we are given the curve Cq with 0(2) structure H generated by e1 =
(x, y) = (0, 0). We wish to find a Weierstrass equation for the quotient curve
Cq=H. Given an elliptic curve C, a Weierstrass equation for C is generated by
choosing a function x1 with a pole of order 2 at the identity and a function y1
with a pole of order 3 at the identity. We need to find such functions x1 and
y1 for C=H.
21
Define the functions x1 and y1 on Cq=H as follows. Given a point P 2 Cq,
define
x1(P )= x(P ) + xt(P )
y1(P )= y(P ) + yt(P ).
Here the functions xt and yt are defined by
xt(P )= x(P + e1)
yt(P )= y(P + e1)
where e1 is the generator of the 0(2) structure of Cq and + denotes addition
in the elliptic curve. It is immediate that the functions x1 and y1 descend to
the quotient C=H.
One may use the formulas for the group law of a Weierstrass curve given in
[45] to obtain
q4
xt= __
x
q4y
yt= -____.
x2
We warn the reader that the formulas of [45] must be modified slightly since
those formulas are for an elliptic curve in Weierstrass form
Ca : y2 + a1xy + a3y = x3 + a2x2 + a4x + a6
whereas our curves
Cq : y2 = 4x(x2 + q2x + q3)
have a coefficient of 4 instead of 1 in front of the x3.
One then checks that the new coordinate functions x1 and y1 satisfy the Weier-
strass equation
y21= 4x31+ 4q2x21- 16q4x1 - 16q2q4. (1.5.1)
The canonical tangent vector associated to the Weierstrass equation 1.5.1 is
the image of the tangent vector of Cq. The right-hand side of equation 1.5.1 has
a canonical root x = -q2. The curve Cq=H with this 0(2) structure e1 = -q2
corresponds to the dual isogeny to Cq ! Cq=H. We must put the Weierstrass
equation 1.5.1 into the form of Cq0 for some tuple q0. This is accomplished, as
seen in Section 1.3.2, by application of the transformation x 7! x - q2. The
curve transforms to
Cq0: y2 = 4x(x2 - 2q2x - 4q4 + q22).
We conclude that the induced map
__ __
_*d: B ! B
22
is given by
_*d(q2)= -2q2
_*d(q4)= -4q4 + q22.
One sees immediately that relation 1.1.2
(_d)2 = _[2]: M10(2) ! M10(2)
is satisfied. Since _[2]is an isomorphism, it follows that _d is as well.
1.5.4 The map OEq
Relation 1.1.1 forces us do define OEq to be the composite
_d 1 OEf 1
OEq : M10(2) -! M0(2) -! M .
Therefore, on Hopf algebroids, the map
__ __ __
OE*q: (B , __B) ! (B , B)
is given by
OE*q(q2)= -2q2
OE*q(q4)= -4q4 + q22
OE*q(r)= 0.
1.6 Extended automorphism groups
In this section we explain how to extend both the automorphism group of
an elliptic curve, and its associated formal group, by the Galois group. We
then explicitly determine the extended automorphism group of the unique
supersingular elliptic curve over F9, and the extended automorphism group of
its formal group.
In characteristic p, there are many different morphisms which all deserve to be
called the Frobenius morphism, and this can lead to confusion. We first recall
some standard material to clarify which one of these versions of the Frobenius
we want to employ.
Let q be a power of p. Let X = X0 xspec(Fq)spec(Fqn) be a (formal) scheme
over Fqn which is obtained from a (formal) scheme X0 over Fq by base change.
Let ss be the projection morphism
ss : X ! spec(Fqn).
23
Let Frobq : Fqn ! Fqn be the qth power Frobenius automorphism, the genera-
tor of the Galois group Gal(Fqn=Fq). Associated to Frob qare several different
Frobenius morphisms on X:
o Frob q: the qth Frobenius on the base.
o Frob relq: the qth relative Frobenius.
o Frob totq: the qth total Frobenius.
These are given by the following diagram.
X @XXXXXXXPPP
@@PP XXXXXXXFrobtotqXXrel
@@@FrobqPPP XXXXXXXXXX
@@@ PP(( _________XXXXXX,,//_
ss@@@@ X Frobq X
@@@ ss|| ss||
@@__fflffl| fflffl|
spec Fqn __Frobq__//_specFqn
The square in the above diagram is a pullback square. If X0 = spec (A) is
affine, then X is the affine scheme spec(A Fq Fqn), and Frob q(respectively
Frob relq) is the qth Frobenius on Fqn (respectively on A), while Frob totqis t*
*he
Frobenius on A Fq Fqn.
Let Aut (X) denote the automorphism group of X, regarded as a (formal)
scheme over spec(Fqn) There is an induced action of Frob q on ff 2 Aut (X)
given by the following pullback.
Frob*qff
X _________//_X
Frobq|| |Frobq|
fflffl| fflffl|
X ____ff___//_X
This action of Frob q gives rise to an action of Gal(Fqn=Fq) on Aut (X). We
define the extended automorphism group Aut =Fq(X) to be the semidirect prod-
uct
Aut =Fq(X) = Aut (X) o Gal(Fqn=Fq)
associated to this action. The group Aut =Fq(X) consists of automorphisms of
the (formal) scheme X which do not cover the identity on spec(Fqn).
At the prime 3 there is one isomorphism class of supersingular elliptic curve
over F9, admitting the Weierstrass presentation
C : y2 = x3 - x.
For the remainder of this section we will focus our attention on the com-
putation of the endomorphism ring End (C), and the automorphism groups
24
Aut =F3(C) and_Aut =F3(C^ ). We shall see that all of the endomorphisms of C
defined over F3 are defined over F9.
Let F = Frobrel3be the relative Frobenius endomorphism of C. The map F is
given by
F : (x, y) 7! (x3, y3).
Define automorphisms t, s 2 Aut (C) by
t :(x, y) 7! (!4x, !6y)
s :(x, y) 7! (x + 1, y).
Here, ! 2 F9 is a primitive 8th root of unity. Explicit computation, using the
formulas for the group law on C [45], demonstrates that we have relations
F 2 = -3 (1.6.1)
t2 = -1 (1.6.2)
F t= -tF (1.6.3)
1
s = __(1 + F ). (1.6.4)
2
From the definitions of the maps, we deduce the following additional relations.
s3 = 1 (1.6.5)
st = ts2 (1.6.6)
Proposition 1.6.1 The automorphism group of C is the group G12 = C3oC4
of order 12 generated by s and t.
Proof. Relations (1.6.5), (1.6.2), and (1.6.6) demonstrate that s and t gener-
ate a subgroup G12 of Aut (C) of order_12 isomorphic to C3 o C4. The group
of automorphisms of C defined over F3 is of order 12 [45, A.1.2]. We therefore
deduce that all of the automorphisms of C are defined over F9, and that they
are contained in G12.
Proposition 1.6.2 The endomorphism ring of C (over F9) is a maximal or-
der of the quaternion algebra
Q=(F 2= -3, t2 = -1, F t = -tF ).
A Z-basis of the order O = End (C) is given by
{s, t, ts, tF }.
__
Proof. Since C supersingular, the ring of endomorphisms defined over F 3
End _F3(C) is a maximal order of a rational quaternion algebra D ramified at
25
the places p and 1. By Proposition 5.1 of [37], D admits the presentation
D = Q*=(i2 = -3, j2 = -1, ij = -ji).
By Proposition 5.2 of [37], there is a maximal order O of D with Z-basis
1 1
{__(1 + i), j, __(j + ji), ji}.
2 2
Let R be the subring of End (C) generated by t, s, and F . Since D is a division
algebra, the algebra map
D ! R Q
sending i to F and j to t is an isomorphism. We deduce that under this
isomorphism, the ring R is a maximal order of D contained in End _F3(C).
Since the order End (C) is maximal, we see that R = End _F3(C). Since the
generators of R are defined over F9, we see that all of the endomorphisms of
C are defined over F9.
The endomorphism ring End (C^ ) of the formal group C^ is the p-completion
of End (C^ ). This follows from a theorem of Tate (see [48]), which states that
the p-completion of the endomorphism ring of an abelian variety is the the
endomorphism ring of its p-divisible group. However, in the case of the elliptic
curve C, this may be deduced in a more explicit manner.
Recall that the endomorphism ring of a height 2 formal group over Fp2 is the
unique maximal order Op of the Qp-division algebra Dp of invariant 1=2 [39,
Appendix B]. The ring Op admits the presentation
Op = W*~~=(S2 = p, woeS = Sw)
where W = W(Fp2) is the Witt ring of Fp2, w ranges over the elements of W,
and oe is the lift of the Frobenius Frob pon Fp2.
Since C is supersingular, the formal group C^ is a height 2 formal group. The
formal group law C^ is seen, using the formulas of [45], to have 3-series
[3]C^ (T ) = -T 9.
The Witt ring W is the ring of integers of the unramified quadratic extension
of Q3. We may therefore identify the Witt ring W with the subring Z3[t] of
O Z3. Let ! 2 F9 be a primitive 8th root of unity. We shall also let !
represent the Teichm"uller lift to W. Relation 1.6.3 implies that in O Z3 we
have
F ! = !3F. (1.6.7)
26
Define S to be the element !-1 F 2 O Z3. Relations 1.6.7 and 1.6.1 imply
that we have
S2 = 3 (1.6.8)
S! = !3S. (1.6.9)
The following proposition follows immediately.
Proposition 1.6.3 There is an isomorphism of W-algebras
O Z3 = End (C) Z3 = End (C^ ) ! O3
given by sending F to !S.
The Morava stabilizer group S = S2 is the group of units of O3
Aut (C^ ) = Ox3.
To understand the extended automorphism groups
G24 = Aut =F3(C) = Aut (C) o Gal
G = Aut =F3(C^ ) = Aut (C^ ) o Gal
we simply must understand the action of the Galois group Gal = Gal(F9=F3).
Let oe = Frob 3denote the generator of Gal. The Galois action on End (C) is
governed by the equations
oe*F = F
oe*t= -t.
We deduce that the Galois action on Aut (C) is given by
oe*s= s
oe*t= -t.
The Galois action on W Aut (C^ ) is given by the lift of oe. The Galois action
on Aut (C^ ) is then determined by the action of oe on S. We have
oe*S = !-2 S.
Remark 1.6.4 This formula for the Galois action on S differs from that that
is in more common usage (see, for instance, [11]). The reason for this is that
the Morava stabilizer group S is usually taken to be the group of automorphisms
of the Honda height 2 formal_group F2 over Fp2. Since the formal groups F2
and C^ are isomorphic over F 3, they have isomorphic automorphism groups.
However, because they are not isomorphic over F9, their automorphism groups
have non-isomorphic Galois actions.
27
1.7 Effect on Morava modules
In this section we summarize the effects of our operations on E2-homology.
This is an adaptation of the techniques of Mahowald and Rezk [30]. Through-
out this section, we freely use the notation established in Section 1.6. We sha*
*ll
denote the Morava E-theory spectrum E2 by E. We are taking the spectrum
E to be the Goerss-Hopkins-Miller E1 ring spectrum associated to the formal
group C^ over F9. When we write the E-homology group E*X, we mean the
Morava module ss*(LK(2)(E ^ X)).
Recall [12] that for a closed subgroup F G we have an isomorphism of
Morava modules
E*EhF ~= Map c(G=F, E*).
Let C be the supersingular elliptic curve of Section 1.6. There is a lift of th*
*is
curve to a curve Ce over E0 = W[[u1]].
eC: y2 = 4x3 + u1x2 + 2x.
Proposition 1.7.1 The formal group law Ce^ is a universal deformation of
C^ .
Proof. We just need to see that the universal map classifying our deformation
induces an isomorphism on the base ring E0. This follows from the fact that
the 3-series of Ce^ is given by [41]
[3]eC^(T ) u1T 3+ 2(1 + u21)T 9+ . . .(mod 3).
Remark 1.7.2 The Serre-Tate theorem gives an equivalence between the cat-
egory of deformations of C and the category of deformations of the formal
group C^ . Because Ce^ is a universal deformation of C^ , we may deduce that
Ce is a universal deformation of C.
We define finite subgroups of G to be the subgroups generated by the following
elements.
G24 = ~~~~
SD16 =
D8 =
Work of Hopkins-Miller [40] and Goerss-Hopkins [17] shows that the G action
on E* lifts to an E1 action on the spectrum E. We have the following spectra
28
as homotopy fixed point spectra.
EhG24 = T MF
EhSD16 = E(2)
EhD8 = T MF0(2)
The relationship to some of these groups and the curve C is given below.
G = Aut =F3(C^ )
G24 = Aut =F3(C)
D8 = Aut =F3(C, H).
The subgroup H is the level 2 structure generated by the point (0, 0) of C.
Remark 1.7.3 We pause to comment on the relationship between T MF and
EO2. The pair (E, eC) is an E1 elliptic spectrum [19]. This E1 elliptic spec-
trum structure is classified by an E1 ring map
~ : T MF ! E.
Because Ce is a universal deformation, the action of G24 on C extends to an
action of G24 on Ce (covering the action of G24 on E*). The classifying map
~ therefore lifts to an E1 ring map
T MF ! EhG24 = EO2.
This map is a K(2)-local equivalence. The essential point is that the K(2)-
localization of T MF will be given as the sections of the sheaf Oellover a form*
*al
neighborhood of the unique supersingular point C of the (3-local) moduli stack
M (see Section 1.2.1).
Given an endomorphism
OE : C ! C
we get an induced endomorphism
ObE: C^ ! C^ .
Assuming that bOEis an automorphism, we may regard bOEas an element of G,
thus giving an E1 ring map
bOE: E ! E.
The induced map E*bOEon E-homology is given by
bOE
E*E _____E*_____//_E*E
|| ||
|| ||
|| ||
Map c(G, E*)_R*_//_Mapc(G, E*)
bOE
29
where RbOEis right multiplication by bOE. If bOEis in the normalizer NG F of a *
*closed
subgroup F , then there is an induced map
ObE: EhF ! EhF .
Being in the normalizer implies that right multiplication descends to the right
coset spaces G=F . Thus the induced map on E-homology is given by
R*bOE
Map c(G=F, E*) -! Map c(G=F, E*).
We may apply this discussion to translate our maps OEf, _d, and _[2]into the
language of homotopy fixed point spectra. The map OE*fis clearly the map
induced from inclusion 'D8 : D8 ,! G24 giving
OE*f= ResG24D8: EhG24 ! EhD8 .
Consider the quotient isogeny
OEH : C ! C=H
where H is the level 2 structure on C generated by the point (x, y) = (0, 0).
The computations of Section 1.5.3 indicate that the curve C=H is given by
the Weierstrass equation
C=H : y2 = x3 + x.
Note that this differs from our Weierstrass equation for C, but since there is
only one isomorphism class of supersingular curve at p = 3, this curve must
be isomorphic to C. Indeed, we find it convenient to take the isomorphism
~!-1 : C=H ! C
given by
x 7! !-2 x
y 7! !-3 y.
Thus we may consider the composite
OEH ~!-1
_H : (C, H) -! C=H --- ! C.
The induced map _bH on the formal group is readily computed from the for-
mulas of Section 1.5.4. One finds that, regarding _bH as an element of G, we
have
_bH = 1 + t.
30
The group element 1 + t is in the normalizer NG D8. Thus the map _*d :
T MF (2) ! T MF (2) is given by
_*d= [1 + t] : EhD8 ! EhD8 .
Since _[2]corresponds to the second power isogeny, it is given by the element
2 2 G, so that
_[2]= [2] : EhG24 ! EhG24.
We could have used 1 - t 2 G to represent the map _*d. It is irrelevant on
EhD8 since the cosets (t + 1)D8 and (1 - t)D8 are equal in G=D8. The formula
(t + 1)(1 - t) = 2
in G implies relation 1.1.2
(_*d)2 = _[2]: EhD8 ! EhD8
required in the construction of Q(2) given in Section 1.2.2.
1.8 Calculation of V (1)*(Q(2)) at p = 3
Let V (1) be the 3-primary Smith-Toda complex. In this section we will com-
pute the V (1)-homology group V (1)*(Q(2)). We shall compare this computa-
tion to V (1)*(S), where S is the (K(2)-local) sphere.
1.8.1 Computation of the ANSS
We shall compute the Adams-Novikov E2 term for Q(2) ^ V (1) using the
hypercohomology spectral sequence 1.4.1 (modulo the ideal (3, v1)). We first
describe the E1 term.
We recall [41] that the the [3]-series of the formal group law of the elliptic
curve Cq has
v1 q2 (mod 3).
Therefore, the_ANSS_for T MF ^ V (1) is the cohomology of the quotient Hopf
algebroid of (B , __B) (see Section 1.4) given by
__
(B =I2, __B=I2)
__
where I2 is the invariant regular ideal (3, q2) B . The ANSS E2 term for
T MF ^ V (1) is computed [4], [41] to be
E2(T MF ^ V (1)) = H*( __B=I2) = F3[q41 , fi] E[ff]
31
where the generators have bidegrees
|q4|= (0, 8)
|fi|= (2, 12)
|ff|= (1, 4).
We have v2 = -q24. The ANSS E2 term for T MF0(2) ^ V (1) is concentrated
on the zero line, and is given by
E2(T MF0(2) ^ V (1)) = F3[q41 ].
The differentials in the complex C*(Q(2) ^ V (1)) of Section 1.4 are computed
from the formulas of Section 1.5. Namely, we have
OE*f(qk4) (q4)k (mod I2)
_*[2](qk4) qk4 (mod I2)
_*d(qk4) (-q4)k (mod I2)
OE*q(qk4) (-q4)k (mod I2).
The non-zero differentials Di in the total complex C*(Q(2) ^ V (1)) are given
by the following formulas.
D0 : C0( __B)! C1( __B) C0( __B) MF0(2)
D0(qk4) (0, 0, qk4) (mod I2), k 1 (mod 2)
__0 2 __1
D1 : C1( __B) C ( __B) MF0(2) ! C ( __B) C ( __B) MF0(2)
D1(0, 0, qk4) (0, 0, -qk4) (mod I2), k 0 (mod *
*2)
D1(0, qk4, 0) (0, 0, -qk4) (mod I2), k 2 Z
We therefore have the following proposition.
Proposition 1.8.1 The E2-term of the ANSS for Q(2) ^ V (1) is given by
F3[fi, v21 ] E[i]{1, ff, h1, b1}.
The generators are given in the following table. The representative is given
the name of the corresponding element of the ANSS E2-term of the layer of
the tower (The brackets refer to the level of the tower that the generator lives
in).
32
_____________________________________________
| |
Generator || Representative ||Bidegree
______________________________________________
| |
fi || fi[0] ||(2, 12)
| 2 |
v2 || -q4[0] ||(0, 16)
| |
ff || ff[0] ||(1, 4)
| |
h1 || ffq4[0] ||(1, 12)
| 3 |
b1 || fiq4[0] ||(2, 36)
| |
i ||1MF [1] - 1MF0(2)[1] ||(1, 0)
_____________________________________________
The differentials in the ANSS for Q(2) ^ V (1) follow from the differentials in
the ANSS for T MF ^ V (1) (see [4], [41]).
Proposition 1.8.2 The differentials in the ANSS for Q(2) ^ V (1) are given
by
d5(vk2)= vk-22fi2h1, k 2, 3, 4, 6, 7, 8 (mod 9)
d9(vk2ff)= vk-32fi5, k 3, 4, 8 (mod 9)
d9(vk2h1)= fi4vk-42, k 3, 7, 8 (mod 9)
d5(vk2b1)= vk2fi3ff, k 0, 1, 2, 5, 6, 7 (mod 9)
and these differentials are propagated freely by multiplication by fi and i. Th*
*is
completely describes the ANSS.
1.8.2 The V (1)-homology groups V (1)*(Q(2)) and V (1)*(S)
Define patterns A, B, and C of homotopy groups as follows
A = X F3{1, b4, v2}
B = Y F3{1, v2} Z
C = Z F3{v42, v52} Y F3{v52}
where X, Y , and Z are the following graded F3-vector spaces.
X = F3{1, ff, fi, fffi, fi2, , fi3, , fi4}
Y = F3{1, ff, fi, fffi, fi2, fffi2, v2h1, fi3, fiv2h1, fi4}
Z = F3{h1, v-12b1, fih1, fiv-12b1, , fi2v-12b1, ,
fi3v-12b1, }
Figure 1.1 gives a graphical description of the patterns A, B, and C, and the
patterns X, Y , and Z sit inside of them as indicated. In this figure, the lines
33
Fig. 1.1. Graphical depiction of patterns A, B, and C.
of the appropriate length depict multiplication by ff, multiplication by fi, and
application of the Toda bracket .
Let Q0(2) be the fiber of the unit
j
Q0(2) ! S -! Q(2).
With these patterns, we have the following computations. The first two com-
putations are given in [15]. The computation of V (1)*(Q(2)) is an immediate
consequence of Proposition 1.8.2.
V (1)*(T MF ) = (A 72A) P [v29 ]
V (1)*(S) = ((B C) 29(B C)_) P [v29 ] E[i]
V (1)*(Q(2)) = (B C) P [v29 ] E[i]
V (1)*(Q0(2)) = 29(B C)_ P [v29 ] E[i]
The computation of the V (1)-homology of Q0(2) follows from the fact that
the computations of the V (1)-homology of S and Q(2) imply there is a short
exact sequence
0 ! V (1)*(Q0(2)) ! V (1)*(S) ! V (1)*(Q(2)) ! 0
Thus it appears that V (1) is built out of Q(2) ^ V (1) and a 28th suspension
of the Brown-Comenetz dual of Q(2) ^ V (1). This is the key computation
to our proof that Q0(2) ' D(Q(2)) (Theorem 2.0.1). Gross-Hopkins duality
[20] implies that after smashing with V (1), Spanier-Whitehead duality and
34
Brown-Comenetz duality agree up to a suitable suspension in the K(2)-local
category.
Part 2: The 3-primary K(2)-local sphere
Let K denote K(2), E denote E2, and Q denote Q(2). The (K-local) Spanier-
Whitehead dual of X will be denoted by
DX = F (X, LK S).
Recall that the nth monochromatic layer MnX is the fiber
MnX ! LnX ! Ln-1X.
The Gross-Hopkins dual InX is the Brown-Comenetz dual of MnX. We shall
let IK X denote the K-localization of InX. In particular, we shall write IK for
IK S. The spectrum IK is invertible [20], [26]. Therefore, we have [46]
IK X ' DX ^ IK . (2.0.1)
In this part of the paper we wish to prove the following theorem.
Theorem 2.0.1 There is a cofiber sequence
Dj j
DQ -! S -! Q
where j is the unit of the ring spectrum Q.
By equation 2.0.1, we have DQ ' IK Q ^ I-1K. We are able to use this equation
to give an explicit description of DQ. We shall prove
Theorem 2.0.2 The spectrum DQ refines a tower of spectra of the form
46T MF0(2) ! 46(T MF _ T MF0(2)) ! 46T MF.
It will turn out that the spectra T MF and T MF0(2) are Gross-Hopkins self-
dual, up to a suitable suspension. This will follow from the relationship of the
Gross-Hopkins dual to the Mahowald-Rezk dual. The Mahowald-Rezk duals
of certain fp-spectra are easily computed using the theory of [30]. The number
46 in Theorem 2.0.2 arises from 46 = 22 + 22 + 2 where the first two 22's arise
from the equations
IK T MF ' 22T MF (Proposition 2.4.1)
I-1K^ T MF ' 22T MF (Proposition 2.5.1)
35
and the final factor of 2 arises from the fact that the tower
T MF ! T MF _ T MF0(2) ! T MF0(2)
has length 2 (Lemma 2.6.2). Thus we recover, at least in form, the tower of
[16]. The maps in the tower for DQ given in Theorem 2.0.2 are the Spanier-
Whitehead duals of the maps in the tower for Q.
We shall first give a general abstract framework for duality pairings in trian-
gulated closed symmetric monoidal categories, where we define the notion of a
Lagrangian for a hyperbolic pairing. We shall prove that for the duality pair-
ing for the K(2)-local sphere is hyperbolic, in the sense that S_is_built from
1
two dual spectra. The Lagrangian for this decomposition will be S = EhG ,
where G1 is the kernel of the reduced norm G1 ! Z3. The decomposition we
consider is well known [12], but this particular interpretation of it is new. We
then use this decomposition to prove that Q is also a Lagrangian for S, thus
proving that S may also be built out of Q and DQ (Theorem 2.0.1).
The second part of this paper is organized as follows. In Section 2.1, we give *
*our
abstract duality framework. In Section 2.2, it is recalled that V (1)-homology
equivalences are K(2)-local equivalences. In Section 2.3 we prove Proposi-
tion 2.3.1, which says that_for_arbitrary n, the canonical pairing on LK(n)S is
hyperbolic with Lagrangian S . Section 2.4 identifies IK T MF and Section 2.5
identifies IK ^ T MF . In Section 2.6 we use these_identifications to prove The-
orem 2.0.2._In_Section 2.7 we define_a spectrum Q , and in Section 2.9 we
prove that Q is a Lagrangian for S . This uses some decompositions of map-
ping spaces of homotopy fixed point spectra that are recalled from [16] in__
Section 2.8. Section 2.10 describes how Q may be built from two copies of Q .
Theorem 2.0.1 is finally proved in Section 2.11.
2.1 Duality pairings
In this section we briefly outline the abstract framework in which the duality
phenomena of this paper reside. Compare with [32], [13].
We first provide some motivation. Suppose that V is a finite dimensional vector
space over a field k. A (bilinear) pairing is a linear homomorphism
ff : V V ! k.
It is a perfect pairing if the adjoint homomorphism
fef: V ! V *
is an isomorphism. A Lagrangian for such a pairing is a subspace W V such
36
that the induced sequence
* *
W -'!V ~= V *-'! W
is a short exact sequence, where the isomorphism between V and V *is given
by eff. A pairing for which a Lagrangian exists is a hyperbolic pairing. We wish
to provide a framework of hyperbolic pairings in the setting of triangulated
closed symmetric monoidal categories.
Let (C, ^, F (-, -), S) be a closed symmetric monoidal category with compat-
ible triangulated structure [33]. Let DX denote the dual F (X, S) of X. One
says that X is reflexive if the natural map
effl: X ! DDX
is an isomorphism. This should be contrasted with the stronger notion of X
being dualizable, where one insists that the natural map
DX ^ X ! F (X, X)
be an isomorphism.
We shall refer to any map
ff : X ^ X ! I
where I is an element of P ic(C), as a pairing on X. A pairing ff will be called
perfect if the adjoint map
fef: X ! F (X, I) ~=DX ^ I
is an isomorphism. Note that it is immediate that if X possesses a perfect
pairing ff, then X is reflexive.
Suppose that X has a perfect pairing ff : X ^ X ! I. Given a map l : Y ! X,
we may define a dual map l_ by the composite
l_ : X -eff!~DX ^ I -Dl^1--!DY ^ I.
=
We shall say that l : Y ! X is a left Lagrangian for ff if the sequence
_
Y -l!X -l! DY ^ I
extends to a distinguished triangle in C. Alternatively, given a map r : X ! Z,
we may define a map r_ to be the composite
-1
r_ : DZ ^ I -Dr^1--!DX ^ I -eff-!~X
=
37
We shall say that r : X ! Z is a right Lagrangian for ff if the sequence
_ r
DZ ^ I -r! X -! Z
extends to a distinguished triangle of C. We have the following proposition
which enumerates some useful facts regarding these definitions.
Proposition 2.1.1 Suppose that X has a perfect pairing ff : X ^ X ! I.
(1) If Y is reflexive, then given l : Y ! X, we have l__ ~= l under the
isomorphism D(DY ^ I) ^ I ~= Y .
(2) If Z is reflexive, then given r : X ! Z, we have r__ ~= r under the
isomorphism D(DZ ^ I) ^ I ~= Z.
(3) If l : Y ! X is a left Lagrangian, then Y is reflexive, and l_ : X ! DY ^I
is a right Lagrangian.
(4) If r : X ! Z is a right Lagrangian, then Z is reflexive, and r_ : DZ ^I !
X is a left Lagrangian.
Proof. We shall only prove (3). The other parts use similar methods. By
hypothesis, there exists a map ffi so that the top row of the following diagram
is a distinguished triangle.
ffi *
* _
___Y_________l________//_______________________________________________*
*___Xl//_DY/^/I_ Y
_______________________________ |
________|ffl1|_______________________________________________________~=eff|*
*||||||ffl1
_________fflffl|___________________________fflffl||| fflffl|
effl___________________________________________D(DY_^/I)/^_IDX/^/I_DY-^1I//_ *
*D(DY ^ I) ^ I
_____________________________________DlD^1l^1 D ffi^1
_______________________________________________~|
_________________________=|
_''__________________________fflffl|
DDY
The second row is distinguished, since it is what one gets when one applies the
functor D(-) ^ I to the first row, after we have turned the triangle once. The
sign on the map D -1ffi ^ 1 is correct: we get one factor of (-1) from turning
the triangle and one factor of (-1) from axiom TC2 of [33]. One concludes
that ffl1 is an isomorphism, hence efflis.
We conclude with a degenerate example.
Example. Given an element I of P ic(C), there is a unique pairing
* ^ * ! I
which is perfect. Suppose that * ! Z is a right Lagrangian. Then there exists
a map ffi so that the sequence
DZ ^ I ! * ! Z -ffi! DZ ^ I
38
is a distinguished triangle, which means that there is an isomorphism
eff= ffi : Z ! DZ ^ I.
The adjoint
ff : Z ^ Z ! I
is a perfect pairing on Z. Conversely, given Z with a perfect pairing taking
values in I, we see that the unique map * ! Z is a right Lagrangian. The
choice of perfect pairing on Z coincides with the choice of how to complete
the sequence DZ ^ I ! * ! Z to a distinguished triangle. Indeed, given that
Z admits a perfect pairing, the collection of perfect pairings on Z is (non-
canonically) isomorphic to Aut (Z), which is also the automorphism group of
the triangle
-1Z ! * ! Z -=!Z.
2.2 K(2)-local equivalences
In what follows we will often want to deduce certain maps are K(2)-local
equivalences. Of course, these are precisely maps which are K(2)-homology
equivalences. However, it is sometimes useful to use homotopy group calcula-
tions instead. A K(1) version of the following lemma was used to identify the
J spectrum with LK(1)S in [6].
Lemma 2.2.1 Suppose that f : X ! Y is a map for which the induced map
f ^ V (1) : X ^ V (1) ! Y ^ V (1)
induces an isomorphism of homotopy groups. Then f is a K(2)-equivalence.
Proof. Since f ^ V (1) is an equivalence, f ^ V (1) ^ E(2) is an equivalence.
But V (1) ^ E(2) is equivalent to K(2). Thus f is a K(2) equivalence.
2.3 LK(n)S is hyperbolic
In this section we work at an arbitrary prime p, and prove that the canonical
duality pairing for the K(n)-local sphere is hyperbolic. For the purposes of
this section everything is implicitly K(n)-local. We abbreviate K for K(n)
and E for En. Let S be the nth Morava stabilizer group. There is a reduced
norm map
S ! Zp
39
whose kernel we shall identify_as as S1. Let G1 denote the Galois extension
1
S1 o Gal(Fpn=Fp)._Let S be the homotopy fixed point spectrum EhG_. The
spectrum S is an E1 ring spectrum [17]. The group Zp acts on S via E1
maps. Let o be a topological generator of Zp. For convenience of group-ring
notation, we shall use multiplicative notation for the additive group Zp. In
[12] it is proved that there is a homotopy fiber sequence
__ ffi r __ o-1 __
-1S -! S -! S --! S . (2.3.1)
Define a pairing fl by the composite
__ __~ __ ffi1
fl : S ^ S -! S -! S (2.3.2)
__
where ~ is the product on S . We shall refer to its adjoint as efl.
__ __
fel: S ! DS
The following proposition_simultaneously states that fl is a perfect pairing,
and that the map S ! S is a (right) Lagrangian for S.
Proposition 2.3.1 The map eflis an equivalence which makes the left square
of the following diagram commute.
___ffi_//__r__//_
-1S S S
ef'l|| |||| ||||
fflffl|_ || _||_
DS __Dr__//S_r__//S
In particular, the bottom row is a cofiber sequence.
For a spectrum X and a profinite set K = lim-iKi, let X[[K]] be defined to
be the homotopy inverse limit
X[[K]] = lim-X ^ (Ki)+ .
i
Note that the natural map
X ^ S[[K]] ! X[[K]]
is not in general an equivalence.
We shall first need the following theorem, which is a proven in [5].
Theorem 2.3.2 Suppose that H and K are closed subgroups of G. Then there
is an equivalence
ff : (E[[G=H]])hK -'! F (EhH , EhK ).
40
Corollary 2.3.3 Suppose that N is a closed normal subgroup of G. Then
there is an equivalence
ff : EhN [[G=N]] '-!F (EhN , EhN ).
Corollary 2.3.4 Suppose that N is a closed normal subgroup of G. Then
there is an action map in the K(n)-local stable homotopy category
, : S[[G=N]] ^ EhN ! EhN
with the property that for every coset gN 2 G=N, the composite
, hN
{gN}+ ^ EhN ! S[[G]] ^ EhN -!E
coincides with the action of gN on EhN .
Proof. Let effbe the adjoint of the map ff of Corollary 2.3.3
eff: EhN [[G=N]] ^ EhN ! EhN .
Then the map , is given by the composite
j^1^1 hN hN effhN
, : S[[G=N]] ^ EhN --- ! E [[G=N]] ^ E -!E
where j is the unit of the ring spectrum EhN .
Lemma 2.3.5 For any closed subgroup H of G, and profinite set K = lim-iKi
for which all of the maps in the tower {Ki} are surjections, the natural map
S[[K]] ^ EhH ! [[K]]EhH
is an equivalence.
The proof of Lemma 2.3.5 will requires the following finiteness result.
Lemma 2.3.6 There exists a dualizable spectrum Y so that
(1) Y -homology isomorphisms are equivalences.
(2) For every closed subgroup H of G, the Y -homology groups Y*EhH are
finite in each degree.
Proof. Let M be a type n complex and let U be an open normal subgroup
of G of finite cohomological dimension (such a subgroup exists since G is
compact p-adic analytic). Let Y be the spectrum EhU ^ M. The spectrum Y
is dualizable since the spectrum EhU is dualizable. In fact, the spectrum EhU
is self-dual [42].
41
Proof of (1). It suffices to show that for any spectrum X, if Y ^ X is null,
then X must be null. In [42] it is shown that the norm map is an equivalence
N : (EhU )hG=U -'!(EhU )hG=U .
Combined with the fact that EhU is self-dual, we have
(Y ^ X)hG=U ' F (EhU , X)hG=U ^ M
' F ((EhU )hG=U , X) ^ M
' F ((EhU )hG=U , X) ^ M
' F (EhG , X) ^ M
' M ^ X.
Therefore M ^ X is null, but this implies that X is null.
Proof of (2). Using Theorem 2.3.2, and the fact that EhU is dualizable and
self-dual, we have
Y ^ EhH = EhU ^ M ^ EhH
' F (EhU , EhH ) ^ M
' (E[G=U])hH ^ M.
As a left H-set, the finite G-set G=U breaks up into a finite set of H-orbits
a
G=U ~= H=(H \ U).
G=UH
Since H=(H \ U) is finite, there is an H-equivariant equivalence
`
E[G=U] ' Map (H=(H \ U), E).
G=UH
By Shapiro's lemma (see, for example, [5]), we deduce that the H-homotopy
fixed points are given by
`
(E[G=U])hH ' EhH\U .
G=UH
The homotopy fixed point spectral sequence gives a spectral sequence
M
Es,t2= Hsc(H \ U; EtM) ) Yt-s(EhH ).
G=UH
The group H \ U, being a subgroup of U, has finite cohomological dimension
[47, 4.1.2], so the spectral sequence has a horizontal vanishing line. Since M
is a type n complex, the E-homology of M is finite in each degree. Since the
group H \ U is a closed subgroup of the p-adic analytic group G, we may
42
conclude that the cohomology groups Hsc(H \ EtM) are finite for each s and
t (combine Theorem 5.1.2, Lemma 5.1.4, and Proposition 4.2.2 of [47]). We
deduce that the Y -homology Y*EhH is finite in each degree.
Proof of Lemma 2.3.5. It suffices to show that the map
S[[K]] ^ EhH ^ Y ! [[K]]EhH ^ Y
is an equivalence, where Y is the dualizable spectrum of Lemma 2.3.6. The
homotopy fixed point spectrum EhH is a colimit of homotopy fixed point
spectra
EhH = lim-! EhV
H V oG
where the groups V are open [12]. The spectra EhV are dualizable [26]. Since
homotopy inverse limits commute with smash products with dualizable spec-
tra, we are reduced to showing that the natural map
lim-!lim([Ki]EhV ^ Y ) ! lim lim ([Ki]EhV ^ Y )
H V oG -i -i H V-! oG
is an equivalence. By Theorem 3.7 of [35], it suffices to show that the induced
maps
lim-! limsss*(EhV ^ Y )[Ki] ! lims lim ss*(EhV ^ Y )[Ki] (2.3.3)
H V oG -i -i H -!V oG
are isomorphisms. Since the inverse systems are Mittag-Leffler, we only need
to investigate s = 0. By Lemma 2.3.6, the homotopy groups ss*(EhV ^ Y ) and
ss*(EhH ^ Y ) are finite in each degree. The map 2.3.3 for s = 0 is therefore t*
*he
composite of the following sequence of isomorphisms.
lim-! lim ss*(EhV ^ Y )[Ki]= lim ss*(EhV ^ Y )[[K]]
H V oG -i H -!V oG
~= lim (ss (EhV ^ Y ) Z[[K]])
H -!V *
oG
~=( lim ss (EhV ^ Y )) Z[[K]]
H -!V *
oG
~=ss*(EhH ^ Y ) Z[[K]]
~=ss*(EhH ^ Y )[[K]]
~=lim ss (EhH ^ Y )[K ]
-i * i
~=lim lim ss (EhV ^ Y )[K ]
-i H -!V * i
oG
43
Corollary 2.3.7 The action map
, : S[[G=N]] ^ EhN ! EhN
of Corollary 2.3.4 lifts to an action map
, : [[G=N]]EhN ! EhN .
We shall be employing Corollary 2.3.7 in the case of_N_ = G1. The group
1
G=G1 = Zp acts on the homotopy fixed point spectrum S = EhG . We shall
need an additional lemma before proceeding to the proof of Proposition 2.3.1.
Lemma 2.3.8 Let X be a p-complete spectrum, and let o be the canonical
topological generator of the profinite group Zp. Then the sequence
[o]-1 ffl
X[[Zp]] --- ! X[[Zp]] -! X
is a cofiber sequence.
Proof. The map ffl is the homotopy inverse limit of the fold maps
X ^ Z=pn + ! X.
Since the composite is null, it will suffice to show that the sequence induces a
short exact sequence on homotopy groups. Since the inverse system
{ss*(X ^ Z=pn +)} = {ss*(X)[Z=pn]}
is Mittag-Leffler, applying ss* gives the sequence
[o]-1
ss*(X)[[Zp]] --- ! ss*(X)[[Zp]] ! ss*(X).
This sequence is right exact for any X, and since ss*(X) is p-complete, this
sequence is left exact.
Proof of Proposition 2.3.1 By Corollary 2.3.3, we have a canonical equiv-
alence __ __ __
ff : S[[Zp]] '-!F (S , S)
__ 1
using the fact that S = EhG and G=G1 ~=Zp. Using the action of the Morava
stabilizer group, given an element ~ 2 Zp, we get an induced map
__ __
[~] : S ! S.
44
The effect of the post- and pre-composition maps under ff are given respec-
tively by
__ __
[~]* =L~ : S[[Zp]] ! S[[Zp]]
__ __
[~]* =R~ : S[[Zp]] ! S[[Zp]]
where L~ is the diagonal left action of ~ and R~ us the right action on the Zp
factor.
We shall first prove that eflis an equivalence. Define a map
__ __ __
efl1: S[[Zp]] ! F (S , S)
whose adjoint is the pairing given by the composite
__ __, __ __~ __
fl1 : S[[Zp]] ^ S -! S ^ S -! S.
__
Here , is given by allowing the Zp to act diagonally on the two S 's (using
the action maps of Corollaries 2.3.7 and 2.3.4), and ~ is the product. We
emphasize that efl1is a different map from ff.
Consider the following diagram:
__ Ro-1 __ ffl __
S [[Zp]]_____//S[[Zp]]_______//S (2.3.4)
' efl1||(1) ' efl1||(2) efl||
_fflffl|__ fflffl|___ fflffl|_
F (S , S)[o]*-1//_F (S_,fS)fi*//_F (S , S1)
The bottom row is a cofiber sequence, since it arises from the cofiber se-
quence 2.3.1. The top row is a cofiber sequences by Lemma 2.3.8. It will
follow that eflis an equivalence once we show that efl1is an equivalence and we
show that squares (1) and (2) of Diagram 2.3.4 commute.
Define a map fi to be the composite
__ __ ,L __
fi : S[[Zp]] -! S [[Zp x Zp]] -! S[[Zp]]
__
where is the diagonal and ,L is given by letting the first Zp act on S through
the map of Corollary 2.3.7. The map fi is an isomorphism, with inverse given
by the composite
__ __ __ ,L __
S[[Zp]] -! S [[Zp x Zp]] -! S[[Zp]]
__
where : Zp ! Zp x Zp is the modified diagonal given by
__ -1
(~) = (~ , ~).
(Here, as always, we are using multiplicative notation for the additive group
Zp.) Then the map efl1is an equivalence because it is the composite of the two
45
equivalences __ __ __ __
efl1: S[[Zp]] '-!S [[Zp]] '-!F (S , S).
fi ff
Square (1) of Diagram 2.3.4 commutes since it is given by the following com-
posite, where each of the squares clearly commutes.
__ Ro-1 __
S [[Zp]]_____//S[[Zp]]
fi|| fi||
__ fflffl|L -1_ fflffl|
S [[Zp]]____o//_S[[Zp]]
ff|| ff||
_fflffl|__ fflffl|___
F (S , S)[o]*-1//_F (S , S)
Square (2) of Diagram 2.3.4 commutes since the adjoint maps form the fol-
lowing commutative diagram.
__ __ ffl^1 __ __
S[[Zp]] ^MS________________//_S^_S_________
_______MMM_______________________________________________*
*___________________________________________________________________
__,__||___~[[Id]]MM___________________________~||_________*
*_________________________
______fflffl|__MM&&______________________fflffl|___________*
*____________________
fl1____________________________________fl____________________*
*______(3)S[[Zf]]fl//__
______________________S^ S S _____________________ *
* p
_________________________qqqq ______________________
_________________________________________________________*
*_______________________~|,qqffi|(4)
________________________|qqq| __________________________*
*____________________
_##_________________________f________________flffl|xxqq*
*qfflffl|_//_1
S ffi S
__
The map , is the action map of Zp on S given by_Corollary 2.3.7. The map
~[[Id]] is the product on the two factors of S. The only portions of the diagram
which don't obviously commute are (3) and (4). Portion (3) commutes since
Zp acts by maps of E1 ring spectra. Since the composite given in the cofiber
sequence
__ [o]-1__ ffi1
S --- !S -!S
is null, ffi coequalizes the Zp-action. Therefore, portion (4) commutes.
We are left with showing that the left-hand square of the diagram in the
statement of the proposition commutes. It suffices to check on adjoints, and
this is accomplished with the following commuting diagram.
__ _1^r//__ __
S ^|SJJJJ S ^ S
| JJJJJJJ ~|
| JJJJJ|
| JJJJfflffl|_
ffi|| S
| |
| ffi|
fflffl| fflffl|
S1 _________S1
46
2.4 Identification of IK(2)T MF
In this section we will prove the following proposition.
Proposition 2.4.1 The spectra T MF and T MF0(2) are Gross-Hopkins self
dual up to the suspensions given below.
IK T MF ' 22T MF
IK T MF0(2) ' 22T MF0(2)
IK E(2) ' 22E(2)
For the remainder of this section, we shall work in the 3-local stable homotopy
category, not localized at K(2).
Recall from [30] that an fp-spectrum is a spectrum whose Fp-cohomology
is finitely presented over the Steenrod algebra A. Let CfnX be the fiber of
the finite localization map X ! LfnX. Mahowald and Rezk [31] studied a
dualization functor Wn defined by
WnX = ICfnX.
Their theory indicated that this dual (which we shall refer to as the Mahowald-
Rezk dual) is quite computable for certain fp-spectra of type n. In particular,
they show that both tmf and BP <2> are self-dual.
Proposition 2.4.2 (Mahowald-Rezk [31]) There are equivalences
W2tmf ' 23tmf
W2BP <2> ' 23BP <2>.
Since tmf0(2) is equivalent to BP <2> _ 8BP <2>, and is periodic of order 8,
we have the following corollary.
Corollary 2.4.3 The Mahowald-Rezk dual of tmf0(2) is given by an equiva-
lence W2tmf0(2) ' 23tmf0(2).
We would like to use these computations to identify the Gross-Hopkins duals
of T MF and T MF0(2). To this end we investigate the relationship between
the Gross-Hopkins dual and the Mahowald-Rezk dual. A spectrum X is said
to satisfy the E(n)-telescope conjecture if the natural map
tn : LfnX ! LnX
is an equivalence. Our interest in such spectra stems from the following propo-
sition.
47
Proposition 2.4.4 Suppose that X is a spectrum which satisfies the E(n)
and E(n - 1)-telescope conjectures. Then there is a cofiber sequence
Wn-1X ! WnX ! InX.
Proof. Applying Verdier's axiom to the composite X ! LnX ! Ln-1X gives
a cofiber sequence
-1MnX ! CnX ! Cn-1X.
Since X satisfies the E(n)-telescope conjecture, we may regard this cofiber
sequence as
-1MnX ! CfnX ! Cfn-1X.
The proposition follows after taking the Brown-Comenetz dual.
In [31], following a suggestion of Hal Sadofsky, it is pointed out that the
chromatic tower computations of [38] imply the following proposition.
Proposition 2.4.5 The spectra BP satisfy the E(m)-telescope conjecture
for each m.
Proof. The spectrum BP satisfies the E(m)-telescope conjecture (see [30,
7.2] and [39, Theorem 6.2]). Since the localization functors Lfm and Lm are
smashing, the E(m)-telescope conjecture holds for any spectrum which may
be obtained from BP by means of iterated cofibers and filtered homotopy
colimits. In particular, the E(m)-telescope conjecture holds for the spectrum
BP .
Since tmf0(2) is equivalent to the wedge BP <2> _ 8BP <2>, one has the fol-
lowing corollary.
Corollary 2.4.6 The spectrum tmf0(2) satisfies the E(m)-telescope conjec-
ture at the prime 3 for every m.
The following lemma will allow us to bootstrap ourselves up from tmf0(2) to
tmf.
Lemma 2 Let T be the 3-cell complex S0 [ff1e4 [ff1e8. Then there is an
equivalence
tmf ^ T ' tmf0(2).
Proof. The spectrum tmf0(2) is a tmf-algebra through the map that forgets
the 0(2) structure
OE*f: tmf ! tmf0(2).
48
Since the (3-primary) homotopy groups of tmf0(2) ' BP <2> _ 8BP <2> are
concentrated in even degrees, the Hurewitz images of the attaching maps of
T in ss*(tmf0(2)) are null, and hence the map OE*ffactors to give a map
eOE*
f: tmf ^ T ! tmf0(2).
The Adams-Novikov E2-term of tmf ^ T , and the induced map eOE*fon Adams-
Novikov E2-terms, are easily computed with an Atiyah-Hirzebruch spectral
sequence, from which we deduce that the map OeE*finduces an isomorphism
on the level of Adams-Novikov E2-terms. We conclude that the map eOE*fis an
equivalence.
From this, we deduce the following.
Proposition 2.4.7 The spectrum tmf satisfies the E(m)-telescope conjecture
at the prime 3 for every m.
Proof. In [39], the complex T is used extensively in the following manner.
One has the cofiber sequences given below.
S0 ! T ! 4Cff1
Cff1 ! T ! S8
Here Cff1 is the cofiber of ff1. These splice together to give a BP <2>-Adams
resolution for tmf.
tmf ^ S0 oo___tmf ^ 3Cff1 oo____tmf ^ S10 oo____tmf ^ 13Cff1 oo___. . .
| | | |
| | | |
fflffl| fflffl| fflffl| fflffl|
tmf0(2) 3tmf0(2) 10tmf0(2) 13tmf0(2)
Let Xi be the ith term of the tower. The tower converges in the sense that
there is an equivalence lim-Xi ' *. Let Yi be given by the following cofiber
sequences
Xi ! X0 ! Yi.
Then by Verdier's axiom there exist cofiber sequences
Yi ! Yi-1 ! N tmf0(2).
Since localization respects cofiber sequences, the spectra Yi inductively satis*
*fy
the E(m)-telescope conjecture. We may recover tmf by
tmf ' lim-Yi.
49
Localization does not respect homotopy inverse limits in general, but this
inverse limit is nice, because the composite
S10 ! 3Cff1 ! S0
is the element fi1 of ss10(S). Since fi61is null in the stable stems, our tower*
* Xihas
the property that Xi+14! Xiis null. This makes the tower Yipro-equivalent to
a constant tower, and thus localizations commute with this homotopy inverse
limit. We deduce that tmf satisfies the E(m)-telescope conjecture.
We recall the following theorem from [26, Theorem 6.19], which also appeared
without proof in [20].
Theorem 2.4.8 ([26], [20]) For any spectrum X, the monochromatic fiber
MnX satisfies
MnX ' MnLK(n)X.
Proof of Proposition 2.4.1. Let K denote K(2). We shall prove that
IK T MF ' -1LK W2tmf. The result will then follow from Proposition 2.4.2.
The arguments for tmf0(2) and BP <2> are similar. Applying LK to the cofiber
sequence given by Proposition 2.4.4, we get a cofiber sequence
LK W1tmf ! LK W2tmf ! IK tmf.
We have an equivalence LK tmf ' LK T MF . Theorem 2.4.8 implies that there
is an equivalence IK tmf ' IK T MF . We are left with proving that the map
LK W2tmf ! IK T MF
is an equivalence. By Lemma 2.2.1, it suffices to show that this map is an
isomorphism on V (1)-homology. This is readily seen from computation. By
Proposition 2.4.2 we have
V (1)*(LK W2tmf) ~= 23V (1)*(T MF ).
Since L1V (1) ' *, we also have IK V (1) ' ILK V (1), and therefore
V (1)*(IK T MF ) ~= 7ss*(I(V (1) ^ T MF )) ~= 23V (1)*T MF
The first isomorphism follows from the fact that the CW-complex V (1) is a
Spanier-Whitehead self-dual complex of dimension 6. The second isomorphism
follows from the computation
V (1)*(T MF ) ~= 56Hom (V (1)*(T MF ), Q=Z)
50
coupled with the fact that V (1)*(T MF ) is periodic of order 72. The map
LK W2tmf ! IK T MF is readily seen to induce an isomorphism between
these V (1)-homology groups.
2.5 Identification of I-1K(2)^ T MF
We now resume the convention that everything is implicitly localized at K =
K(2). The aim of this section is to prove the following proposition.
Proposition 2.5.1 There are equivalences
IK ^ T MF ' -22T MF
IK ^ T MF0(2) ' -22T MF0(2)
IK ^ E(2) ' -22E(2)
Proof. The Gross-Hopkins duality theorem [20], [46] states that there is an
isomorphism
E*(IK ) ~= 2E*[det]
of Morava modules. Here [det] indicates a twisting with the determinant char-
acter. There is a homotopy fixed point spectral sequence
Hs(G24; Et(IK )) ) sst-s(T MF ^ IK ). (2.5.1)
The E2 term of this spectral sequence is thus identified by the Gross-Hopkins
duality theorem. Since G24 is contained in the kernel of the determinant, we
have an isomorphism
H*(G24; E*(IK )) ' H*(G24; 2E*)
The spectral sequence 2.5.1 is a spectral sequence of modules over the spectral
sequence of algebras
Hs(G24; Et) ) sst-s(T MF ) (2.5.2)
which computes the homotopy groups of T MF . The structure of the latter
spectral sequence forces the spectral sequence 2.5.1 to be isomorphic to the
spectral sequence 2.5.2 up to a suspension congruent to 2 modulo 24. Thus
there exists a map
SN ! T MF ^ IK
which is detected on the 0-line of spectral sequence 2.5.1 which extends (using
the T MF -module structure of T MF ^ IK ) to an equivalence
N T MF -'!T MF ^ IK .
51
We just need to determine the value of N, which is congruent to 2 modulo
24 and which is only unique modulo 72, the order of periodicity of T MF .
Unfortunately, we are unable to avoid appealing to [15] to determine N. This
computation was summarized in Section 1.8.2. It implies that
ss*(V (1)) ~=ss*( 28IV (1)) ~=ss*( 22IK ^ V (1))
which forces the value of N to be -22.
The identification of IK ^ T MF0(2) is similar. The spectrum T MF0(2) has
the homotopy fixed point model EhD8 . The group D8 is in the kernel of the
determinant, and the ANSS for IK ^ EhD8 is concentrated on the zero line, so
there are no differentials. Thus one has an isomorphism
H*(D8; E*IK ) ~=H*(D8; 2E*)
which gives an equivalence
IK ^ T MF0(2) ' 2T MF0(2) ' -22T MF0(2)
using the eightfold periodicity of T MF0(2).
The case of E(2) is slightly different. The spectrum E(2) is given by the homo-
topy fixed point spectrum EhSD16, but the subgroup SD16 is not contained in
the kernel of the determinant. In fact, we have an isomorphism of the restricted
determinant
det #GSD16~=O
where O is the non-trivial character of SD16=D8, regarded as an SD16 repre-
sentation. Using the fact that there is an isomorphism of SD16-modules
E* ZO3~= 8E*
it follows that there are isomorphisms
H*(SD16; E*IK ) ~=H*(SD16; 2E*[det]) ~=H*(SD16; 10E*)
which realize to equivalences
IK ^ E(2) ' 10E(2) ' -22E(2)
using the 16-fold periodicity of E(2).
2.6 Proof of Theorem 2.0.2
We begin with the an identification of DT MF , DT MF0(2), and DE(2).
52
Proposition 2.6.1 There are equivalences
DT MF ' 44T MF
DT MF0(2) ' 44T MF0(2)
DE(2) ' 44E(2).
Proof. Since IK is invertible, we have for any X [46]
DX ^ IK ' IK X.
Smashing this equation with I-1Kgives
DX ' IK X ^ I-1K.
In particular, we may let X be T MF , T MF0(2), or E(2), and then apply
Propositions 2.4.1 and 2.5.1.
We shall make use of the following lemma, which follows from the equivalence
of fibers and cofibers (up to a suspension) in the stable homotopy category.
Lemma 2.6.2 Suppose X is a spectrum which refines the tower
A0 ! A1 ! . .!.Ad.
Then DX is a spectrum which refines the dual tower
dDAd ! dDAd-1 ! . . .dDA0.
Since Q refines a tower of the form
T MF ! T MF _ T MF0(2) ! T MF0(2),
Theorem 2.0.2 follows immediately from Proposition 2.6.1 and Lemma 2.6.2.
__
2.7 A spectrum which is half of S
In the next few sections we work our way toward_proving Theorem 2.0.1.
1
Proposition 2.3.1 in particular implies_that S = EhG is self dual. In this
section_we introduce a spectrum Q which will turn out to be a Lagrangian for
S .
Following [16, Sec. 4], let O be the the sign representation for SD16=D8 re-
garded as an SD16-representation. (See Section 1.7 for descriptions of these
53
subgroups.) There is a splitting
T MF0(2) ' E(2) _ 8E(2)
which realizes the splitting
Z3[[G=D8]] ~=Z3[[G=SD16]] ZO3"GSD16
where the SD16-representation O has been induced up to a G-representation.
The first summand is generated by [!] + 1 2 Z3[[G=D8]] and the second by
[!] - 1 2 Z3[[G=D8]]. Here, and elsewhere, we use brackets to denote the
group-like elements of our group rings when confusion may arise as to where
the sums are taking place. Let
: T MF0(2) ! 8E(2)
be the projection map.
One deficiency of our map OE*qis that the corresponding element 1 + t 2 G (see
__ 1
Section 1.7) has norm det(1+t) = 2. Since S = EhG , where G1 is the kernel of
the reduced norm, it would be more natural to consider maps corresponding
to elements of reduced norm 1.
p __
Let 2 2 W be the choice of a square root of 2 which reducespto_!2 in F9.
Such an element exists by Hensel's lemma. We may regard 2 as an element
of G. Define a map __
*
OEq: T MF ! T MF0(2)
which in the language of homotopy fixed point spectra, is given by the com-
posite
'*D8 [(1+t)=p _2]
EhG24 --! EhD8 --- - - - !EhD8 .
p __ p __ p __
(The Frobenius takes 2 to - 2, so one may verify that 2 is an element of
the normalizer NG D8.)
__
Definition 2.7.1 Define Q to be the homotopy equalizer
*
__ __OOEf//_8
Q _____//T MF_____// E(2).
OOE*q
The following lemma implies that we may transfer_our understanding of the
*
effect of OE*qon V (1)-homology to that of OEq.
Lemma 2.7.2 The maps
__*
V (1)*OE*q, V (1)*OEq: V (1)*(T MF ) ! V (1)*(T MF0(2))
are identical.
54
__* p __ p __
Proof. The map OEqdiffers from OE*qby a factor of 2 2 W. Since 2 reduces
to !2 2 F9, and V (1)*T MF0(2)pis_concentrated in degrees congruent to 0
modulo 8, the effect of 2 on V (1)*T MF0(2) is multiplication by !8 = 1.
Let e8 be the generator in degree 8 of ss*( 8E(2)). Then our computations
in Section 1.5 combined with Lemma 2.7.2 imply that when we compute the
effects of the map on Adams-Novikov E2-terms
__* * 8
* O (OEq- OEf) : E2(V (1) ^ T MF ) ! E2(V (1) ^ E(2))
we have 8
__* * k < -v(k-1)=22e8,if k is odd
* O (OEq- OEf)(q4) = :
0 if k is even
__
From these formulas one easily computes the ANSS E2-term E2(Q ). The ANSS
differentials are induced from the differentials in the ANSS for T MF just as
in Proposition 1.8.2. One finds that (using the patterns of homotopy groups
described in Section 1.8) we have an isomorphism
__ 9
V (1)*(Q ) = (B C) P [v2 ]. (2.7.1)
__
This should be compared to the computation of the V (1)-homology of S given
in [15]. __
V (1)*(S ) = ((B C) 29(B C)_) P [v29 ] (2.7.2)
__ __
We see that V (1)*(Q_) is an additive summand of both V (1)*(S_) and V (1)*(Q),
and that V (1)*(S ) is_the_direct sum of a copy of V (1)*(Q ) and a shifted copy
of the dual of V (1)*(Q ).
2.8 Recollections from [16]
In [16, Prop. 2.6] the mapping spectra F (EhF1, EhF2) are described for Fi
closed subgroups of G, with F2 finite. The authors of [16] prove the following
proposition.
Proposition 2.8.1 (Goerss-Henn-Mahowald-Rezk, [16]) We have
Y
F (EhF1, EhF2) ' EhFx
x2F2\G=F1
where Fx is the finite subgroup F2 \ xF1x-1.
The product in Proposition 2.8.1 must be properly interpreted as an appropri-
ate homotopy inverse limit, using the profinite structure of the double coset
space F2\G=F1.
55
We are primarily concerned with the following spectra.
EhG24 = T MF
EhD8 = T MF0(2)
EhSD16 = E(2)
The homotopy groups of EhF are computed in [16] for F finite. We shall use
the following observations about ss*EhF for F finite.
Lemma 2.8.2 Suppose that one of the Fi is G24 and the other is either G24,
D8, or SD16. Then the homotopy of each of the summands ss*(EhFx) described
in Proposition 2.8.1 is concentrated in degrees congruent to 0 modulo 4 and
1, 3, 10, 13, 27, 30 modulo 36.
Proof. The group Fx = F2 \ xF1x-1 is contained in a conjugate of the sub-
group G24, and contains the central element -1, since each of the subgroups
G24, D8, and SD16 contains the element -1. We conclude that either: (1) the
group Fx contains a conjugate of the cyclic group C6 = ~~~~, or (2) the
order of the group Fx is prime to 3, and Fx contains the element -1. In either
case Fx contains the element -1. The ring of invariants EFx*is concentrated
in degrees congruent to 0 modulo 4, since this is true of the invariants E*1
[16, Cor. 3.15]. If we are in case (2), then there is no higher cohomology, and
the homotopy fixed point spectral sequence collapses to give
ss*(EhFx) = EFx*.
If we are in case (1), then the computations of [16, Thm 3.10, Rmk. 3.12]
indicate that elements of ss*(EhFx) arising in the homotopy fixed point spectral
sequence from group cohomology elements of cohomological degree greater
than 0 may only lie in degrees congruent to 1, 3, 10, 13, 27, or 30 modulo
36.
Lemma 2.8.3 Suppose that one of the Fi is either D8 or SD16 and the other
is either G24, D8, or SD16. Then the homotopy of each of the summands
ss*(EhFx) described in Proposition 2.8.1 is concentrated in degrees congruent
to 0 modulo 4.
Proof. Since one of the subgroups Fi has order prime to 3, and both of the
subgroups Fi contain the central element -1, we are in case (2) of the proof
of Lemma 2.8.2
56
__
2.9 A Lagrangian decomposition of S
__ 1 __
Our calculations of the V (1)-homology of S = EhG (2.7.2) and Q (2.7.1)
suggest the following Lagrangian decomposition.
Proposition 2.9.1 There is a map __jsuch that the following is a fiber se-
quence
__ D_j __ _j__
DQ -! S -! Q .
__ __
The map D__jis the dual of the map __j, using the equivalence DS ' -1S given
by Proposition 2.3.1.
Remark 2.9.2 __Proposition 2.9.1 provides an alternative construction of the
resolution of S given in [16].
The inclusion 'G24 : G24 ,! G1 induces a map
__
'*G24: S ! T MF.
__ __*
The spectrum Q is the homotopy equalizer of the maps O OEqand O OE*f
__* p __
(Definition 2.7.1) where OEqand OE*fcorrespond to the elements (t + 1)= 2 and
1 of G, respectively. Both of these elements have norm 1. Therefore the two
composites
_*
____'*G24//_ ___OEq//_ _____//8
S T MF _____//T MF0(2) E(2)
OE*f
actually agree. Thus there is a lift of '*G24to a map
__ __
S ! Q .
This is the map __j.
Remark 2.9.3 The lift __jof_'*G24is unique, since Proposition 2.8.1 implies
there are no nontrivial maps S ! 7E(2).
Lemma 2.9.4 The composite
__ D_j __ _j__
DQ -! S -! Q
is null.
Proof. We shall prove that
__ __
[ DQ , Q] = 0.
57
__ __
Since Q is built from T MF and 7E(2), by Proposition 2.6.1, DQ is built
from 45T MF and 38E(2). Lemma 2.8.2 implies that the following groups
are zero.
ss45(F (T MF, T MF ))= 0 ss38(F (T MF, E(2)))= 0
ss38(F (E(2), T MF ))= 0 ss31(F (E(2), E(2)))= 0
__ __
It follows that there are no essential maps DQ ! Q .
Proof of Proposition 2.9.1 Let F be the fiber of the map __j. Lemma 2.9.4
implies that there exists a lift f making the following diagram commute.
__ _j __
F ___'__//_S_____//_Qaa____OO
______
____|D_j
f _______|
___|
DQ
Consider the maps V (1)*' and V (1)*D__jon V (1)-homology. The map V (1)*' is
an isomorphism onto kerV (1)*__jsince V (1)*__jis surjective. The map V (1)*D__j
is seen to be an isomorphism onto kerV (1)*__jfrom our explicit knowledge of
the V (1)-homology groups. The effect of the map V (1)*D__jon V (1)-homology
is determined from V (1)*__j, since V (1)*D__jis just the Pontryagin dual of
V (1)*__j. Since both V (1)*' and V (1)*D__jinduce isomorphisms onto their im-
ages, V (1)*f must be an isomorphism. By Lemma 2.2.1 the map f must
therefore be an equivalence.
__
2.10 Building Q from Q
In the first part of this section we will produce a map
rQ __
Q -! Q
__
which will turn out to be equivalent to the unit map Q = Q ^ S -1^r-!Q ^ S.
The map rQ will be produced by the methods of [16, Sec. 4]. Specifically, we
will prove the following lemma.
Lemma 2.10.1 There exists a continuous map ae of Z3[[G]]-modules making
the following diagram commute
[(t+1)=p _2]-1 O G
Z3[[G=G24]]oo________Z3[[G=D8]] oo________________Z3 "SD16
__
|| _ae____
|| ____
|| fflffl___
Z3[[G=G24]]oo_______([t+1]-1)_([2]-1)____ Z3[[G=D8]] Z3[[G=G24]]
(2.10.1)
58
We postpone the proof of Lemma 2.10.1.
Lemma 2.10.2 The map ae induces a map ae* making the following diagram
commute. _
O(OE*q-OE*f)8
T MF __________// E(2)
|| OO
|| ae*|
|| |
|| |
T MF d0-d1//_T MF0(2) _ T MF
Proof. By Proposition 2.8.1, for H finite, the mapping space F (EhH , E(2))
is equivalent to a homotopy inverse limit of products of homotopy fixed point
spectra EhG with G contained in SD16. In particular, the groups G have
order prime to 3, and thus the ANSS for the mapping space F (EhH , E(2)) is
concentrated in the zero line.
Therefore, the edge homomorphism gives a sequence of isomorphisms
[EhH , 8E(2)] ~=Hom cZ3[SD16](Z3, ss8(E)[[G=H]])
~=Hom cZ (Z , ZO-1 bZ [[G=H]])
3[SD16]3 3 3
~=Hom cZ (ZO, Z [[G=H]])
3[SD16]3 3
~=Hom cZ (ZO "G , Z [[G=H]]).
3[[G]]3 SD16 3
It follows that the map ae gives rise to the desired map, and the commutativity
of the square in the statement of the lemma follows from the commutativity
of Diagram 2.10.1.
The map of towers
T MF _________//_ 8E(2)_____________//*OO
|| OO
|| ae*| |
|| | |
|| | |
T MF ____//_T MF0(2) _ T MF ____//_T MF0(2)
induces a map __
rQ : Q ! Q .
The importance of this map for us is encoded in the following lemma and its
corollary.
Lemma 2.10.3 There is an equivalence , making the following diagram com-
mute. __
Q _1^j//_DDQ ^ S
__
DDD ______
rQDDDD',____
!!Dfflffl___
Q
59
The proof of Lemma 2.10.3 is postponed.
Corollary 2.10.4 There is a fiber sequence
__ rQ __
-1Q ! Q -! Q .
Proof. Simply smash the fiber sequence
__ ffi r __
-1S -! S -! S
with Q, and apply Lemma 2.10.3.
In order to prove Lemma 2.10.1, we shall need the following technical result.
Lemma 2.10.5 The sequence
f ffl
Z3[[G=D8]] Z3[[G=G24]] -! Z3[[G=G24]] -! Z3 ! 0
is exact, where ffl is the augmentation and f is the map ([t + 1] - 1) ([2] -*
* 1).
Proof. The composite ffl O f is clearly zero. Let N Z3[[G=G24]] be the kernel
of ffl. We must show that f surjects onto N. Lemma 4.3 of [16] implies that it
suffices to show that the reduced map
F3 f : F3 Z3[[G]](Z3[[G=D8]] Z3[[G=G24]])! F3 Z3[[G]]N
is surjective. This is equivalent to showing that
f* : Ext0Z3[[G]](N, F3) ! Ext0Z3[[G]](Z3[[G=D8]] Z3[[G=G24]], F3)
is injective. The group Ext 0(N, F3), and the map f*, may be easily deduced
from our computations of the ANSS E2-terms for V (1)-homology, from which
it is explicitly seen that indeed f* is injective.
Proof of Lemma 2.10.1. For any Z3[[G]] module M, producing a map
ZO3"GSD16! M
is equivalent to specifying an element of M on which SD16 acts on with the
sign representation. Let x 2 Z3[[G=G24]] be the element corresponding to the
map p __
([(t + 1)= 2] - 1) O : ZO3"GSD16! Z3[[G=G24]].
60
The element x is in the kernel of ffl, hence by Lemma 2.10.5 it lifts to an
element exof Z3[[G=D8]] Z3[[G=G24]]. While it is not necessarily true that ex
generates a copy of O, the weighted average
1 X
y = ___ O(g)[g]ex
16 g2SD16
does have the property that Z3y ~=ZO3. The element y corresponds to the map
ae.
__*
Proof of Lemma 2.10.3_We_first must define ,. Note that since_ O_OEq
and O OE*fare maps of S -modules, the homotopy equalizer Q is a S -module.
Therefore there is a right action map
__ __ __
~ : Q ^ S ! Q .
The map , is defined to be the composite
rQ^r __ __~ __
, : Q --- ! Q ^ S -! Q
__
where r is the unit S ! S.
Recall that, in the notation of Section 1.8.2, we have
V (1)*(Q) = (B C) P [v29 ] E[i]
__ 9
V (1)*(Q ) = (B C) P [v2 ].
The definition of rQ implies that diagram below commutes.
rQ __
Q _________//Q
| |
| |
fflffl| fflffl|
T MF ______T MF
Since these are maps of S-modules, the only possibility is for the induced map
V (1)*rQ on V (1)-homology is to be the quotient by the ideal generated by i.
__
In Appendix A, we prove Theorem A.0.2, which says that S has a cell decom-
position
S0 [i e0 [i e0 . ...
__
It follows that V (1)*(Q^S ) is the quotient of V (1)*(Q) by the ideal generated
by i, and the map __
(1 ^ r)* : ss*(Q) ! ss*(Q ^ S)
is the quotient map.
61
Our map , is easily shown to make the diagram of in the statement of
Lemma 2.10.3 commute, and since V (1)*(rQ ) and V (1)*(1 ^ r) are both sur-
jections onto the same image, V (1)*(,) must be an isomorphism. Lemma 2.2.1
implies that , is an equivalence.
2.11 Proof of Theorem 2.0.1
__
In this section we piece together our Lagrangians Q to prove Theorem 2.0.1.
We first observe that the sequence of Theorem 2.0.1 looks like a fiber sequence
on V (1)-homology.
Lemma 2.11.1 Using the notation of Section 1.8.2, there is an isomorphism
of short exact sequences.
V (1)*DQ _____Dj______//_V (1)*S____j_____//V (1)*Q
|| || ||
|| || ||
|| i || j ||
29(B C)_ R _____// (B2C)9(B C)_ R____//(B C) R
where R is the ring P [v29 ] E[i] and the maps in the bottom row are the
obvious inclusions and projections.
Proof. The map j was essentially computed in Section 1.8.2. The map Dj
is, up to suspension, just the Pontryagin dual of the map j.
We shall need the following lemma.
__ __
Lemma 2.11.2 There is an equivalence DQ ^ S ' DQ .
__
Proof. Although S is not dualizable, we will nevertheless show that the nat-
ural map __ __
^ : DQ ^ DS ! D(Q ^ S)
is an equivalence. The equivalence of the statement of the lemma is then the
composite
__ 1^efl __ ^ __ D,-1 __
DQ ^ -1S --!'DQ ^ DS -! D(Q ^ S) ---'! DQ
where eflis the equivalence given in Proposition 2.3.1, and , is the equivalence
given in Lemma 2.10.3.
62
__
Let r : S ! S be the unit. Consider the following commutative diagram.
__ 1^Dr
DQ ^ DS ____//_DQ ^ DS (2.11.1)
^ || ' ^||
fflffl|_ fflffl|
D(Q ^ S) D(1^r)__//_DQ
We claim that when we apply V (1)-homology to Diagram 2.11.1, we get the
following, using the notation of Section 1.8.2
V (1)*1^Dr
(B C) P //_.i____//_(B C) R (2.11.2)
V (1)*^|| |V|(1)*^||
fflffl| .i ||
(B C) P V//(1)*D(1^r)//_(B C) R
where P = P [v29 ] and R = P E[i]. We are claiming that the top and bottom
maps are the inclusions given by multiplication by i. We see that the left hand
map V (1)*^ must therefore be an isomorphism. Therefore, by Lemma 2.2.1,
it is an equivalence, and the lemma is proven.
We are left with showing that the top and bottom maps in Diagram_2.11.2
behave as claimed on V (1)-homology. The cellular model of S given in The-
orem A.0.2 implies that there is an the following isomorphism of short exact
sequences.
__1^ffi 1^r __
0_____//V (1)*(DQ ^ -1S )____//V (1)*(DQ ^ S)_____//V (1)*(DQ ^ S)____//_0
|| || ||
|| || ||
|| || ||
0________//(B C) P____.i___//(B C) R _______//_(B C) P_____//0
(2.11.3)
Smashing the left hand square of the diagram of Proposition 2.3.1 with DQ,
we get a commutative diagram
__ 1^ffi
DQ ^ -1S ____//_DQ ^ S
1^efl'|| ||||
fflffl|_ ||
DQ ^ DS 1^Dr_//_DQ ^ S
from which it follows that 1 ^ Dr behaves as advertised on V (1)-homology.
63
Taking the Spanier-Whitehead dual of the diagram of Lemma 2.10.3 gives the
following commutative diagram.
__ D(1^r)
D(Q ^ S) ____//_DQ
D, '|| ||||
_fflffl|_ ||
DQ __DrQ__//_DQ
Gross-Hopkins duality implies that V (1)-cohomology is, up to a shift in degree,
the Pontryagin dual of V (1)-homology. Therefore, the description of V (1)*(rQ )
given in the proof of Lemma 2.10.3 dualizes to give the desired behavior of
V (1)*D(1 ^ r) in Diagram 2.11.2.
We are now ready to prove_a weaker version of Theorem 2.0.1 where we have
smashed everything with S .
Lemma 2.11.3 The sequence
__Dj^1 __j^1 __
DQ ^ S --- ! S ^ S --! Q ^ S
is a fiber sequence.
__
Proof. The cellular description_of S given in Theorem A.0.2 implies that on
V (1)-homology, smashing with S corresponds to modding out by the ideal
generated by i. Using Lemma 2.11.1, we see that we have an isomorphism of
short exact sequences
__ Dj __ j __
V (1)*DQ ^ S ___________//_V (1)*S________//V (1)*Q ^ S
|| || ||
|| || ||
|| i || j ||
29(B C)_ P _____// (B2C)9(B C)_ P____//(B C) P
where P = P [v29 ].
__ __
By Lemma 2.10.3, we have an equivalence_Q_^ S ' Q , and by Lemma 2.11.2
we have an equivalence DQ ^ S ' DQ . In the proof of Lemma 2.9.4, it was
proven that __ __
[ DQ , Q] = 0.
Thus the composite
__Dj^1 __ j^1 __
DQ ^ S --- ! S --! Q ^ S
__
is null. The induced map from DQ ^ S to the fiber of j ^ 1 is seen to be a
V (1)-homology equivalence, hence by Lemma 2.2.1, it is an equivalence.
64
We will now complete the proof of Theorem 2.0.1. The reason that Theo-
rem 2.0.1 is more difficult to prove than Proposition 2.9.1 is that the compos-
ite
Dj j
DQ -! S -! Q
cannot be shown to be null by dimensional considerations alone. However, we
have_shown in Lemma 2.11.3 that after smashing the above sequence with
S we get_a cofiber_sequence. In fact, smashing the sequence with the map
o - 1 : S ! S, we get a map of fiber sequences
__ Dj^1 __ j^1 __
DQ ^ S _________//_S ^_S_______//_Q ^ S
1^(o-1)| 1^(o-1)| 1^(o-1)|
|fflffl_Dj^1 fflffl|_j^1 fflffl|_
DQ ^ S _________//_S ^_S_______//_Q ^ S
which is good in the sense of [36]. This means the there exist induced maps f,
g, and h below which make a 3 x 3 diagram of fiber sequences, as displayed
below.
__ __ __ __
-2Q ^ S _____// -1DQ ^ S _____// -1S_____// -1Q ^ S (2.11.4)
|| (-1) || || ||
fflffl| fflffl| fflffl| fflffl|
-1Q _____f_____//___________________DQg//__________________________*
*___Sh//_____________________________Q
| | | |
| | | |
fflffl|_ fflffl|_ _fflffl|_ fflffl|_
-1Q ^ S _______//DQ ^ S_________//S_______//_Q ^ S
| | | |
| | | |
fflffl|_ fflffl|_ _fflffl|_ fflffl|_
-1Q ^ S _______//DQ ^ S_________//S_______//_Q ^ S
We must show that g = Dj and h = j. Our identifications of the Spanier-
Whitehead duals of T MF and E(2) in Proposition 2.6.1 in particular_imply
that these spectra are reflexive. Therefore the spectrum Q , being the fiber
of a map between reflexive spectra, is itself reflexive. Given different maps g0
and h0 in place of g and h making Diagram 2.11.4 commute, we see that the
difference g - g0 is in the image of the homomorphism
__
ffi* : [DQ, -1S ] ! [DQ, S].
We have isomorphisms
__ __
[DQ, -1S ] ~=[DQ, DS ] (Proposition 2.3.1)
~=[DQ ^ __S, S]
~=[ D(__Q), S] (Lemma 2.11.2)
~=ss1(DD__Q)
~=ss1(__Q) (__Qis reflexive)
65
Similarly, the difference h - h0 is in the image of the homomorphism
__
(1 ^ ffi)* : [S, -1Q ^ S] ! [S, Q]
and we have, by Lemma 2.10.3, an isomorphism
__ __
[S, -1Q ^ S] ~=ss1(Q ).
__ __
However, ss1(Q ) = 0, since Q is built from T MF and 7E(2). Thus g and h
are uniquely determined up to homotopy having the property that they make
Diagram 2.11.4 commute. Since Dj and j make Diagram 2.11.4 commute,
we must have g = Dj and h = j, as desired. This completes the proof of
Theorem 2.0.1.
__
Appendix A: A cellular model for S
In this appendix we shall always be working in the K(n)-local category at an
arbitrary prime p. Let Gn be the nth extended Morava stabilizer group, and
let G1nbe the kernel of the reduced norm [12]
1 ! G1n! Gn ! Zp ! 1.
__ 1
Let S be the homotopy fixed point spectrum EhGn. Let o 2 Zp be a topological
generator. In [12], the following theorem is proven as an application of the
authors' continuous homotopy fixed point construction.
Theorem A.0.1 (Hopkins-Miller) There is a cofiber sequence
__ ffi r __ [o]-1__
-1S -! S -! S --- ! S.
The element i exists in ss-1(S), and is given by the composite
__ ffi
i : S-1 -r! -1S -! S.
We shall prove the_following theorem, which gives a K(n)-local cellular de-
composition of S . We used_this decomposition in the proof of Lemma 2.10.3
to compute V (1)*(Q(2) ^ S).
Theorem A.0.2 There exist complexes Ciiwith cellular decompositions
Cii= S0 [i e0 [i e0 [i . .[.ie0.
_________-z________"
i
__
There is an equivalence lim-!Cii' S.
66
Remark A.0.3 The existence of the complexes Ciiis equivalent to the the
Toda bracket
~~*
i
being defined and containing zero, for every i.
The remainder of this appendix is dedicated to proving this theorem. Our
models for the intermediate complexes Ciiwill be the homotopy fibers of the
map ([o ] - 1)i, giving fiber sequences
__ ffiii ri __ ([o]-1)i_
-1S -! Ci -! S -- - - !S.
The complex C0iis just the sphere S.
We must explain why the homotopy fibers Ciihave the cellular models as
claimed. We shall inductively prove that there are cofiber sequences
ii-1 i-1 'i i i
S-1 -- ! Ci -! Ci -! S (A.1)
where the map ii-1 will make the following diagram commute.
_ii-1//_i-1
S-1EE Ci
EEE |i-1
iEEEE |
E""fflffl|
S
Thus ii-1 is a lift of i to the whole complex Ci-1i.
The existence of the cofiber sequence A.1 is a direct consequence of Verdier's
axiom.
_________ __________//_________//_
-1S _______ -1S * S (A.2)
| | |
| | | |
([o]-1)i-1| ([o]-1)i | ([o]-1)i-1|
| | |
fflffl|_[o]-1fflffl|_ fflffl| _fflffl|_
-1S _______// -1S___ffi_//S____r____//_S
| | || |
| | || |
ffii-1| ffii| || ffii-1|
| | || |
fflffl| fflffl| || fflffl|
Ci-1i___'i____//_________________Cii/i/______________________*
*_Sii-1//_ Ci-1i
| | || |
ri-1|| ri|| | ri-1||
| | || |
fflffl|_ fflffl|_ fflffl| fflffl|_
S ____________S_________//*_______// S
67
The commutativity of the following diagram implies that the composite of ii
with the projection onto the top cell is i.
_______ii______//_i
S-1FF________________Ci|<*