An overview of abelian varieties in homotopy theory

Abstract

This is an expository article on the theory of formal group laws in homotopy theory, with the goal of leading to the connection with higher-dimensional abelian varieties and automorphic forms. These are roughly based on a talk at the conference "New Topological Contexts for Galois Theory and Algebraic Geometry."

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arXiv:0810.0507v2 [math.AT] 12 Feb 2009
An overview of abelian varieties in homotopy theory
TYLER LAWSON
We give an overviewof the theory of formal grouplaws in homotopytheory,lead-
ing to the connection with higher-dimensional abelian varieties and automorphic
forms.
55P99; 55Q99
1 Introduction
The goal of this paper is to provide an overview of joint work with Behrens on
topological automorphic forms [8]. The ultimate hope is to introduce a somewhat
broad audience of topologists to this subject matter connecting modern homotopy
theory, algebraic geometry, and number theory.
Through an investigation of properties of Chern classes, Quillen discovered a connec-
tion between stable homotopy theory and 1-dimensional formal group laws [41]. After
almost 40 years, the impacts of this connection are still being felt. The stratification of
formal group laws in finite characteristic gives rise to the chromatic filtration in stable
homotopy theory [42], and has definite calculational consequences. The nilpotence
and periodicity phenomena in stable homotopy groups of spheres arise from a deep
investigation of this connection [13].
Formal group laws have at least one other major manifestation: the study of abelian
varieties. The examination of this connection led to elliptic cohomology theories and
topological modular forms, or tmf [25]. One of the main results in this theory is
the construction of a spectrum tmf, a structured ring object in the stable homotopy
category. The homotopy groups of tmf are, up to finite kernel and cokernel, the ring
of integral modular forms [10] via a natural comparison map. The spectrum tmf is
often viewed as a “universal” elliptic cohomology theory corresponding to the moduli
of elliptic curves. Unfortunately, the major involved parties have not yet published a
full exposition of this theory. The near-future reader is urged to consult [5], as well as
seek out some of the unpublished literature and reading lists on topological modular
forms if more background study is desired.

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Tyler Lawson
Algebraic topology is explicitly tied to 1-dimensional formal group laws, and so the
formal group lawsofhigher-dimensional abelian varieties (and larger possible “height”
invariants ofthose) areinitially not connected totopology. Thegoal of[8]wastocreate
generalizations of the theory of topological modular forms, through certain moduli of
abelian varieties with extra data specifying 1-dimensional summands of their formal
group laws.
The author doubts that it is possible to cover all of this background to any degree of
detailwithintheconfinesofapaperofreasonable size, evenrestricting tothose subjects
that are of interest from a topological point of view. In addition, there are existing
(and better) sources for this material. Therefore, our presentation of this material is
informal, and wewilltry tolist references for those who findsome subject ofinterest to
them. We assume a basic understanding of stable homotopy theory, and an inevitable
aspect of the theory is that we require more and more of the language of algebraic
geometry as we proceed.
A rough outline of the topics covered follows.
In sections 2 and 3 we begin with some background on the connection between the
theory of complex bordism and formal group laws. We next discuss in section 4 the ba-
sic theories of Hopf algebroids and stacks, and the relation between stack cohomology
and the Adams-Novikov spectral sequence in section 5. We then discuss the problem
of realizing formal group law data by spectra, such as is achieved by the Landweber
exact functor theorem and the Goerss-Hopkins-Miller theorem, in section 6. Examples
of multiplicative group laws are discussed in section 7, and the theories of elliptic
cohomology and topological modular forms in sections 8 and 9. We then discuss the
possibility of moving forward from these known examples in section 10, by discussing
some of the geometry of the moduli of formal groups and height invariants.
The generalization of the Goerss-Hopkins-Miller theorem due to Lurie, without which
the subject of topological automorphic forms would be pure speculation, is introduced
in section 11. We view it as our point of entry: given this theorem, what kinds of new
structures in homotopy theory can we produce?
The answer, in the form of various moduli of higher-dimensional abelian varieties,
appears in section 12. Though the definitions of these moduli are lifted almost directly
from the study of automorphic forms, we attempt in sections 13, 14, and 15 to indicate
why this data is natural to require in order produce moduli satisfying the hypotheses
of Lurie’s theorem. In section 16, we try to indicate why some initial choices are made
the way they are.

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An overview of abelian varieties in homotopy theory
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One of the applications in mind has been the construction of finite resolutions of the
K(n)-local sphere. Henn has given finite length algebraic resolutions allowing compu-
tation of the cohomology of the Morava stabilizer group in terms of the cohomology of
finite subgroups [22]. Goerss-Henn-Mahowald-Rezk [17] and Behrens [6] gave anal-
ogous constructions of the K(2)-local sphere at the prime 3 out of a finite number of
spectra of the form EhG
of the Morava stabilizer group. The hope is that these constructions will generalize
to other primes and higher height by considering diagrams of abelian varieties and
isogenies.
2, where E2is a Lubin-Tate spectrum and G is a finite subgroup
None of the (correct) material in this paper is new.
2 Generalized cohomology and formal group laws
Associated to a generalized cohomology theory E with (graded) commutative multi-
plication, we can ask whether there is a reasonable theory of Chern classes for complex
vector bundles.
The base case is that of line bundles, which we view as being represented by homotopy
classes of maps X → BU(1) = CP∞for X a finite CW-complex. An orientation of
E is essentially a first Chern class for line bundles. More specifically, it is an element
u ∈ E2(CP∞) whose restriction to E2(CP1)∼= E0is the identity element 1 of the ring
E∗. For any line bundle L on X represented by a map f : X → CP∞, we have an
E-cohomology element c1(L) = f∗(u) ∈ E2(X) which is the desired first Chern class.
Orientations do not necessarily exist; for instance, real K-theory KO does not have an
orientation. When orientations do exist, we say that the cohomology theory is complex
orientable. An orientation is not necessarily unique; given any orientation u, any
power series v =?biui+1with bi∈ E2i,b0= 1 determines another orientation and
another Chern class. Any other orientation determines and is determined uniquely by
such a power series.
Given an orientation of E, we can derive computations of E∗(BU(n)) for all n ≥ 0,
and conclude that for a vector bundle ξ on a finite complex X there are higher Chern
classes ci(ξ) ∈ E2i(X) satisfying naturality, the Cartan formula, the splitting principle,
and almost all of the desirable properties of Chern classes in ordinary cohomology.
See [1].

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Tyler Lawson
The one aspect of this theory that differs from ordinary cohomology has to do with
tensor products. For line bundles L1 and L2, there is a tensor product line bundle
L1⊗ L2formed by taking fiberwise tensor products. On classifying spaces, if Liare
classified by maps fi: X → BU(1), the tensor product is classified by µ ◦ (f1× f2),
where µ: BU(1) × BU(1) → BU(1) comes from the multiplication map on U(1).
Thereisauniversalformulaforthetensorproductoftwolinebundlesin E-cohomology,
given by the formula
c1(L ⊗ L′) =
for ai,j∈ E2i+2j−2. This formula is valid for all line bundles but the coefficients ai,j
depend only onthe orientation. Weoften denote this power series in thealternate forms
?
ai,jc1(L)ic1(L′)j
?
ai,jxiyj= F(x,y) = x +Fy.
This last piece of notation is justified as follows. The tensor product of line bundles is
associative, commutative, and unital up to natural isomorphism, and so by extension
the same is true for the power series x +Fy:
• x +F0 = x,
• x +Fy = y +Fx, and
• (x +Fy) +Fz = x +F(y +Fz).
These can be written out in formulas in terms of the coefficients ai,j, but the third
is difficult to express in closed form. A power series with coefficients in a ring R
satisfying the above identities is called a (commutative, 1-dimensional) formal group
law over R, or just a formal group law.
The formal group law associated to E depends on the choice of orientation. However,
associated to a different orientation v = g(u), the formal group law G(x,y) = x +Gy
satisfies
g(x +Fy) = g(x) +Gg(y).
We say that two formal group laws differing by such a change-of-coordinates for a
power series g(x) = x + b1x2+ ··· are strictly isomorphic. (If we forget which
orientation we have chosen, we have a formal group law without a choice of coordinate
on it, or a formal group.)
The formal group detects so much intricate information about the cohomology theory
E that it is well beyond the scope of this document to explore it well [42]. For certain
cohomology theories E (such as Landweber exact theories discussed in section 6), the
formal group determines the cohomology theory completely. One can then ask, for

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An overview of abelian varieties in homotopy theory
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some spaces X, to understand the cohomology groups E∗(X) in terms of the formal
groupdata. Forexample, if X = BU?6?, thisturnsouttoberelatedtocubical structures
[2].
3 Quillen’s theorem
Thereisacohomology theory MU associated tocomplex bordism thatcomes equipped
with an orientation u. There is also a “smash product” cohomology theory MU ∧ MU
coming equipped with two orientations u and v, one per factor of MU, and hence with
two formal group laws with a strict isomorphism g between them.
The ring L = MU∗ forming the ground ring for complex bordism was calculated
by Milnor [36], and similarly for W = (MU ∧ MU)∗. Both are infinite polynomial
algebras over Z, the former on generators xi in degree 2i, the latter on the xi and
additional generators bi(also in degree 2i). The following theorem, however, provides
a more intrinsic description of these rings.
Theorem 1 (Quillen) The ring L is a classifying object for formal group laws in the
category of rings, i.e. associated to a ring R with formal group law F, there is a unique
ring map φ: L → R such that the image of the formal group law in L is F.
Thering W∼= L[b1,b2,...] isaclassifyingobjectforpairsofstrictlyisomorphicformal
group lawsinthe category ofrings, i.e. associated to aring R withastrict isomorphism
g between formal group laws F and G, there is a unique ring map φ: W → R such
that the image of the strict isomorphism in W is the strict isomorphism in R.
(It is typical to view these rings as geometric objects Spec(L) and Spec(W), which
reverses the variance; in schemes, these are classifying objects for group scheme
structures on a formal affine schemeˆA1.)
The structure of the ring L was originally determined by Lazard, and it is therefore
referred to as the Lazard ring.
There are numerous consequences of Quillen’s theorem. For a general multiplicative
cohomology theory R,thetheory MU ∧ R inheritstheorientation u,andhenceaformal
group law. The cohomology theory MU ∧ MU ∧ R has two orientations arising from
the orientations of each factor, and these two differ by a given strict isomorphism. For
more smash factors, this pattern repeats. Philosophically, we have a ring MU∗R with
formalgrouplaw,togetherwithacompatibleactionofthegroupofstrictisomorphisms.