Duality for Topological Modular Forms

Vesna Stojanoska

Received: May 18, 2011 Revised: November 24, 2011

Communicated by Stefan Schwede

Abstract. It has been observed that certain localizations of the spectrum of topological modular forms are self-dual (Mahowald-Rezk, Gross-Hopkins). We provide an integral explanation of these results that is internal to the geometry of the (compactified) moduli stack of elliptic curves M, yet is only true in the derived setting. When 2 is inverted, a choice of level 2 structure for an elliptic curve provides a geometrically well-behaved cover of M, which allows one to consider T mf as the homotopy fixed points of T mf(2), topological modular forms with level 2 structure, under a natural action by GL_{2}(ℤ/2). As a result of Grothendieck-Serre duality, we obtain that T mf(2) is self-dual. The vanishing of the associated Tate spectrum then makes T mf itself Anderson self-dual.

2010 Mathematics Subject Classification: Primary 55N34. Secondary 55N91, 55P43, 14H52, 14D23. Keywords and Phrases: Topological modular forms, Brown-Comenetz duality, generalized Tate cohomology, Serre duality.

1 Introduction

There are several notions of duality in homotopy theory, and a family of such originates with Brown and Comenetz. In [BC76], they introduced the spectrum I_{ℚ/ℤ} which represents the cohomology theory that assigns to a spectrum X the Pontryagin duals Hom_{ℤ}(π_{−∗}X, ℚ/ℤ) of its homotopy groups. Brown-Comenetz duality is difficult to tackle directly, so Mahowald and Rezk [MR99] studied a tower approximating it, such that at each stage the self-dual objects turn out to have interesting chromatic properties. In particular, they show that self-duality is detected on cohomology as a module over the Steenrod algebra.

Consequently, a version of the spectrum tmf of topological modular forms possesses self-dual properties. The question thus arises whether this self-duality of tmf is already inherent in its geometric construction. Indeed, the advent of derived algebraic geometry not only allows for the construction of an object such as tmf, but also for bringing in geometric notions of duality to homotopy theory, most notably Grothendieck-Serre duality. However, it is rarely possible to identify the abstractly constructed (derived) dualizing sheaves with a concrete and computable object. This stands in contrast to ordinary algebraic geometry, where a few smallness assumptions guarantee that the sheaf of differentials is a dualizing sheaf (eg. [Har77, III.7]). Nevertheless, the case of the moduli stack of elliptic curves^{1} M^{0} and topological modular forms allows for a hands-on approach to Serre duality in a derived setting. Even if the underlying ordinary stack does not admit a traditional Serre duality pairing, the derived version of M^{0} is considerably better behaved. The purpose of this paper is to show how the input from homotopy theory brings duality back into the picture. Conversely, it provides an integral interpretation of the aforementioned self-duality for tmf that is inherent in the geometry of the moduli stack of elliptic curves. The duality that naturally occurs from the geometric viewpoint is not quite that of Brown and Comenetz, but an integral version thereof, called Anderson duality and denoted I_{ℤ} [And69, HS05]. After localization with respect to Morava K-theory K(n) for n > 0, however, Anderson duality and Brown-Comenetz duality only differ by a shift. Elliptic curves have come into homotopy theory because they give rise to interesting one-parameter formal groups of heights one or two. The homotopical version of these is the notion of an elliptic spectrum: an even periodic spectrum E, together with an elliptic curve C over π_{0}E, and an isomorphism between the formal group of E and the completion of C at the identity section. Étale maps Spec π_{0}E → M^{0} give rise to such spectra; more strongly, as a consequence of the Goerss-Hopkins-Miller theorem, the assignment of an elliptic spectrum to an étale map to M^{0} gives an étale sheaf of E_{∞}-ring spectra on the moduli stack of elliptic curves. Better still, the compactification of M^{0}, which we will hereby denote by M, admits such a sheaf, denoted O^{top}, whose underlying ordinary stack is the usual stack of generalized elliptic curves [Beh07]. The derived global sections of O^{top} are called T mf, the spectrum of topological modular forms. This is the non-connective, non-periodic version of T mf.^{2} The main result proved in this paper is the following theorem:

Theorem 13.1. The Anderson dual of T mf[1/2] is Σ^{21}T mf[1/2].

The proof is geometric in the sense that it uses Serre duality on a cover of M as well as descent, manifested in the vanishing of a certain Tate spectrum.

1The moduli stack of elliptic curves is sometimes also denoted M_{ell} or M_{1,1}; we chose the notation from [DR73].

2We chose this notation to distinguish this version of topological modular forms from the connective tmf and the periodic T MF.

Documenta Mathematica 17 (2012) 271–311

Acknowledgments

This paper is part of my doctoral dissertation at Northwestern University, supervised by Paul Goerss. His constant support and guidance throughout my doctoral studies and beyond have been immensely helpful at every stage of this work. I am indebted to Mark Behrens, Mark Mahowald, and Charles Rezk for very helpful conversations, suggestions, and inspiration. I am also grateful to the referee who has made extraordinarily thorough suggestions to improve an earlier version of this paper, and to Mark Behrens and Paul Goerss again for their patience and help with the material in Section 5.

2 Dualities

We begin by recalling the definitions and properties of Brown-Comenetz and Anderson duality. In [BC76], Brown and Comenetz studied the functor on spectra

X ↦ I_{ℚ/ℤ}^{∗}(X) = Hom_{ℤ}(π_{−∗}X, ℚ/ℤ).

Because ℚ/ℤ is an injective ℤ-module, this defines a cohomology theory, and is therefore represented by some spectrum; denote it I_{ℚ/ℤ}. We shall abuse notation, and write I_{ℚ/ℤ} also for the functor

X ↦ F (X, I_{ℚ/ℤ}),

which we think of as “dualizing” with respect to I_{ℚ/ℤ}. And indeed, if X is a spectrum whose homotopy groups are finite, then the natural “double-duality” map X → I_{ℚ/ℤ}I_{ℚ/ℤ}X is an equivalence. In a similar fashion, one can define I_{ℚ}^{3} to be the spectrum corepresenting the functor

X ↦ Hom_{ℤ}(π_{−∗}X, ℚ).

The quotient map ℚ → ℚ/ℤ gives rise to a natural map I_{ℚ} → I_{ℚ/ℤ}; in accordance with the algebraic situation, we denote by I_{ℤ} the fiber of the latter map. As I have learned from Mark Behrens, the functor corepresented by I_{ℤ} has been introduced by Anderson in [And69], and further employed by Hopkins and Singer in [HS05]. For R any of ℤ, ℚ, or ℚ/ℤ, denote also by I_{R} the functor on spectra X ↦ F (X, I_{R}). When R is an injective ℤ-module, we have that π_{∗}I_{R}X = Hom_{ℤ}(π_{−∗}X, R). Now for any spectrum X, we have a fiber sequence

I_{ℤ}X → I_{ℚ}X ⟶^{φ} I_{ℚ/ℤ}X,  (1)

3In fact, I_{ℚ} is the Eilenberg-MacLane spectrum Hℚ.

Documenta Mathematica 17 (2012) 271–311

giving a long exact sequence of homotopy groups

\xymatrix{⋯ & π_{t+1}I_{ℚ/ℤ}X \ar[rr] \ar[dr] &  & π_{t}I_{ℤ}X \ar[rr] \ar[dr] &  & π_{t}I_{ℚ}X & ⋯ \\  &  & Coker π_{t+1}φ \ar[ur] &  & Ker π_{t}φ \ar[ur]}

But the kernel and cokernel of π_{∗}φ compute the derived ℤ-duals of the homotopy groups of X, so we obtain short exact sequences

0 → Ext_{ℤ}^{1}(π_{−t−1}X, ℤ) → π_{t}I_{ℤ}X → Hom_{ℤ}(π_{−t}X, ℤ) → 0.

We can think of these as organized in a two-line spectral sequence:

Ext_{ℤ}^{s}(π_{t}X, ℤ) ⇒ π_{−t−s}I_{ℤ}X.  (2)

Remark 2.1. In this project we will use Anderson duality in the localization of the category of spectra where we invert some integer n (we will mostly be interested in the case n = 2). The correct way to proceed is to replace ℤ by ℤ[1/n] everywhere. In particular, the homotopy groups of the Anderson dual of X will be computed by Ext_{ℤ[1/n]}^{s}(π_{t}X, ℤ[1/n]).

2.1 Relation to K(n)-local and Mahowald-Rezk Duality

This section is a digression meant to suggest the relevance of Anderson duality by relating it to K(n)-local Brown-Comenetz dualty as well as Mahowald-Rezk duality. We recall the definition of the Brown-Comenetz duality functor in the K(n)- local category from [HS99, Ch. 10]. Fix a prime p, and for any n ≥ 0, let K(n) denote the Morava K-theory spectrum, and let L_{n} denote the Bousfield localization functor with respect to the Johnson-Wilson spectrum E(n). There is a natural transformation L_{n} → L_{n−1}, whose homotopy fiber is called the monochromatic layer functor M_{n}, and a homotopy pull-back square

\xymatrix{L_{n} \ar[r] \ar[d] & L_{K(n)} \ar[d] \\ L_{n−1} \ar[r] & L_{n−1}L_{K(n)},}  (3)

implying that M_{n} is also the fiber of L_{K(n)} → L_{n−1}L_{K(n)}. The K(n)-local Brown-Comenetz dual of X is defined to be I_{n}X = F (M_{n}X, I_{ℚ/ℤ}), the Brown-Comenetz dual of the n-th monochromatic layer of X. By (3), I_{n}X only depends on the K(n)-localization of X (since M_{n}X only depends on L_{K(n)}X), and by the first part of the proof of Proposition 2.2 below, I_{n}X is K(n)-local. Therefore we can view I_{n} as an endofunctor of the K(n)-local category.

Note that after localization with respect to the Morava K-theories K(n) for n ≥ 1, I_{ℚ}X becomes contractible since it is rational. The fiber sequence (1) then gives that L_{K(n)}I_{ℚ/ℤ}X = L_{K(n)}ΣI_{ℤ}X. By Proposition 2.2 below, we have that I_{n}X is the K(n)-localization of the Brown-Comenetz dual of L_{n}X. In particular, if X is already E(n)-local, then

I_{n}X = L_{K(n)}ΣI_{ℤ}X.  (4)

In order to define the Mahowald-Rezk duality functor [MR99] we need some preliminary definitions. Let T_{i} be a sequence of finite spectra of type i and let Tel(i) be the mapping telescope of some v_{i} self-map of T_{i}. The finite localization L_{n}^{f} of X is the Bousfield localization with respect to the wedge Tel(0) ∨ ⋯ ∨ Tel(n), and C_{n}^{f} is the fiber of the localization map X → L_{n}^{f}X. A spectrum X satisfies the E(n)-telescope conjecture [Rav84, 10.5] if and only if the natural map L_{n}^{f}X → L_{n}X is an equivalence (eg. [MR99]). Let X be a spectrum whose 𝔽_{p}-cohomology is finitely presented over the Steenrod algebra; the Mahowald-Rezk dual W_{n} is defined to be the Brown-Comenetz dual I_{ℚ/ℤ}C_{n}^{f}. Suppose now that X is an E(n)-local spectrum which satisfies the E(n) and E(n − 1)-telescope conjectures. This condition is satisfied by the spectra of topological modular forms [Beh06, 2.4.7] with which we are concerned in this work. Then the monochromatic layer M_{n}X is the fiber of the natural map L_{n}^{f}X → L_{n−1}^{f}X. Taking the Brown-Comenetz dual of the first column in the diagram of fiber sequences

[figure: Σ^{−1}M_{n}X; ⋆; M_{n}X; C_{n}^{f}X; X; L_{n}^{f}X; C_{n−1}^{f}X; X; L_{n−1}^{f}X]

implies the fiber sequence [Beh06, 2.4.4]

W_{n−1}X → W_{n}X → L_{K(n)}I_{ℤ}X,

relating Mahowald-Rezk duality to K(n)-local Anderson duality. We have used the following result.

Proposition 2.2. For any X and Y, the natural map F (L_{n}X, Y) → F (M_{n}X, Y) is K(n)-localization.

Proof. First of all, we need to show that F (M_{n}X, Y) is K(n)-local. This is equivalent to the condition that for any K(n)-acyclic spectrum Z, the function spectrum F (Z, F (M_{n}X, Y)) = F (Z ∧M_{n}X, Y) is contractible. But the functor M_{n} is smashing, and is also the fiber of L_{K(n)} → L_{n−1}L_{K(n)}, by the homotopy pull-back square (3). Therefore, for K(n)-acyclic Z, M_{n}Z is contractible, and the claim follows. It remains to show that the fiber F (L_{n−1}X, Y) is K(n)-acyclic. To do this, smash with a generalized Moore spectrum T_{n}. This is a finite, Spanier-Whitehead self-dual (up to a suspension shift) spectrum of type n, which is therefore E(n − 1)-acyclic. A construction of such a spectrum can be found in Section 4 of [HS99]. We have (up to a suitable suspension)

F (L_{n−1}X, Y) ∧ T_{n} = F (L_{n−1}X, Y) ∧ DT_{n} = F (T_{n}, F (L_{n−1}X, Y))
= F ((L_{n−1}T_{n}) ∧ X, Y ) = ∗,

implying that F (L_{n−1}X, Y) is K(n)-acyclic, which proves the proposition.

3 Derived Stacks

In this section we briefly recall the notion of derived stack which will be useful to us; a good general reference for the classical theory of stacks is [LMB00]. Roughly speaking, stacks arise as solutions to moduli problems by allowing points to have nontrivial automorphisms. The classical viewpoint is that a stack is a category fibered in groupoids. One can also equivalently view a stack as a presheaf of groupoids [Hol08]. Deligne-Mumford stacks are particularly well-behaved, because they can be studied by maps from schemes in the following sense. If X is a Deligne-Mumford stack, then X has an étale cover by a scheme, and any map from a scheme S to X is representable. (A map of stacks f : X → Y is said to be representable if for any scheme S over Y, the (2-categorical) pullback X ×_{Y}S is again a scheme.)

Derived stacks are obtained by allowing sheaves valued in a category which has homotopy theory, for example differential graded algebras, simplicial rings, or commutative ring spectra. To be able to make sense of the latter, one needs a nice model for the category of spectra and its smash product, with a good notion of commutative rings and modules over those. Much has been written recently about derived algebraic geometry, work of Lurie on the one hand [Lur11b, Lur09, Lur11a] and Toën-Vezossi on the other [TV05, TV08]. For this project we only consider sheaves of E_{∞}-rings on ordinary sites, and consequently we avoid the need to work with infinity categories. Derived Deligne-Mumford stacks will be defined as follows.

Definition 3.1. A derived Deligne-Mumford stack is a pair (X , O_{X}^{top}) consisting of a topological space (or a Grothendieck topos) X and a sheaf O_{X}^{top} of E_{∞}-ring spectra on its small étale site such that

(a) the pair (X , π_{0}O_{X}^{top}) is an ordinary Deligne-Mumford stack, and

(b) for every k, π_{k}O_{X}^{top} is a quasi-coherent π_{0}O_{X}^{top}-module.

Here and elsewhere in this paper, by π_{∗}F of a sheaf of E_{∞}-rings we will mean the sheafification of the presheaf U ↦ π_{∗}F(U). Next we discuss sheaves of E_{∞}-ring spectra. Let C be a small Grothendieck site. By a presheaf of E_{∞}-rings on C we shall mean simply a functor F : C^{op} → (E_{∞}-Rings). The default example of a site C will be the small étale site X_{ét} of a stack X [LMB00, Ch. 12]. A presheaf F of E_{∞}-rings on C is said to satisfy hyperdescent or that it is a sheaf provided that the natural map F(X) → holim F(U_{•}) is a weak equivalence for every hypercover U_{•} → X. Hyperdescent is closely related to fibrancy in Jardine’s model category structure [Jar87, Jar00]. Specifically if F → QF is a fibrant replacement in Jardine’s model structure, then [DHI04] shows that F satisfies hyperdescent if and only if F (U) → QF (U) is a weak equivalence for all U. When the site C has enough points, one may use Godement resolutions in order to “sheafify” a presheaf [Jar87, Section 3]. In particular, since X_{ét} has enough points we may form the Godement resolution F → GF. The global sections of GF are called the derived global sections of F and Jardine’s construction also gives a spectral sequence to compute the homotopy groups

H^{s}(X , π_{t}F) ⇒ π_{t+s}RΓF.  (5)

4 Moduli of Elliptic Curves and Level Stuctures

In this section, we summarize the results of interest regarding the moduli stacks of elliptic curves and level structures. Standard references for the ordinary geometry are Deligne-Rapoport [DR73], Katz-Mazur [KM85], and Silverman [Sil86]. A curve over a base scheme (or stack) S is a proper, flat morphism p : C → S, of finite presentation and relative dimension one. An elliptic curve is a curve p : C → S of genus one whose geometric fibers are connected, smooth, and proper, equipped with a section e : S → C or equivalently, equipped with a commutative group structure [KM85, 2.1.1] . These objects are classified by the moduli stack of elliptic curves M^{0}. The j-invariant of an elliptic curve gives a map j : M^{0} → 𝔸^{1} which identifies 𝔸^{1} with the coarse moduli scheme [KM85, 8.2]. Thus M^{0} is not proper, and in order to have Grothendieck-Serre duality this is a property we need. Hence, we shall work with the compactification M of M^{0}, which has a modular description in terms of generalized elliptic curves: it classifies proper curves of genus one, whose geometric fibers are connected and allowed to have an isolated nodal singularity away from the point marked by the specified section e [DR73, II.1.12]. If C is smooth, then the multiplication by n map [n] : C → C is finite flat map of degree n^{2} [DR73, II.1.18], whose kernel we denote by C[n]. If n is invertible in the ground scheme S, the kernel C[n] is finite étale over S, and étale locally isomorphic to (ℤ/n)^{2}. A level n structure is then a choice of an isomorphism ϕ : (ℤ/n)^{2} → C[n] [DR73, IV.2.3]. We denote the moduli stack (implicitly understood as an object over Spec ℤ[1/n]) which classifies smooth elliptic curves equipped with a level n structure by M(n)^{0}. In order to give a modular description of the compactification M(n) of M(n)^{0}, we need to talk about so-called Néron polygons. We recall the definitions from [DR73, II.1]. A (Néron) n-gon is a curve over an algebraically closed field isomorphic to the standard n-gon, which is the quotient of ℙ^{1} × ℤ/n obtained by identifying the infinity section of the i-th copy of ℙ^{1} with the zero section of the (i + 1)-st. A curve p : C → S is a stable curve of genus one if its geometric fibers are either smooth, proper, and connected, or Néron polygons. A generalized elliptic curve is a stable curve of genus one, equipped with a morphism + : C^{sm} ×_{S} C → C which

(a) restricts to the smooth locus C^{sm} of C, making it into a commutative group scheme, and

(b) gives an action of the group scheme C^{sm} on C, which on the singular geometric fibers of C is given as a rotation of the irreducible components.

Given a generalized elliptic curve p : C → S, there is a locally finite family of disjoint closed subschemes (S_{n})_{n≥1} of S, such that C restricted to S_{n} is isomorphic to the standard n-gon, and the union of all S_{n}’s is the image of the singular locus C^{sing} of C [DR73, II.1.15]. The morphism n : C^{sm} → C^{sm} is again finite and flat, and if C is an mgon, the kernel C[n] is étale locally isomorphic to (μ_{n} × ℤ/(n, m)) [DR73, II.1.18]. In particular, if C a generalized elliptic curve whose geometric fibers are either smooth or n-gons, then the scheme of n-torsion points C[n] is étale locally isomorphic to (ℤ/n)^{2}. These curves give a modular interpretation of the compactification M(n) of M(n)^{0}. The moduli stack M(n) over ℤ[1/n] classifies generalized elliptic curves with geometric fibers that are either smooth or ngons, equipped with a level n structure, i.e. an isomorphism φ : (Z/n)^{2} → C[n] [DR73, IV.2.3]. Note that GL_{2}(ℤ/n) acts on M(n) on the right by pre-composing with the level structure, that is, g : (C, φ) ↦ (C, φ ∘ g). If C is smooth, this action is free and transitive on the set of level n structures. If C is an n-gon, then the stabilizer of a given level n structure is a subgroup of GL_{2}(ℤ/n) conjugate to

the upper triangular matrices U = \{(\begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix})\}.

The forgetful map M(n)^{0} → M^{0}[1/n] extends to the compactifications, where it is given by forgetting the level structure and contracting the irreducible components that do not meet the identity section [DR73, IV.1]. The resulting map q : M(n) → M[1/n] is a finite flat cover of M[1/n] of degree |GL_{2}(ℤ/n)|. Moreover, the restriction of q to the locus of smooth curves is an étale GL_{2}(ℤ/n)-torsor, and over the locus of singular curves, q is ramified of degree n [Kat73, A1.5]. In fact, at the “cusps” the map q is given as

q : M(n)^{∞} ≅ Hom_{U}(GL_{2}(ℤ/n), M^{∞}) → M^{∞}.

Note that level 1 structure on a generalized elliptic curve C/S is nothing but the specified identity section e : S → C. Thus we can think of M as M(1). The objects M(n), n ≥ 1, come equipped with (ℤ-)graded sheaves, the tensor powers ω_{M(n)}^{∗} of the sheaf of invariant differentials [DR73, I.2]. Given a generalized elliptic curve p : C ⇄ S : e, let I be the ideal sheaf of the closed embedding e. The fact that C is nonsingular at e implies that the map p_{∗}(I/I^{2}) → p_{∗}ω_{C/S} is an isomorphism [Har77, II.8.12]. Denote this sheaf on S by ω_{C}. It is locally free of rank one, because C is a curve over S, with a potential singularity away from e. The sheaf of invariant differentials on M(n) is then defined by

ω_{M(n)}(S ⟶^{C} M(n)) = ω_{C}.

It is a quasi-coherent sheaf which is an invertible line bundle on M(n). The ring of modular forms with level n structures is defined to be the graded ring of global sections

MF (n)_{∗} = H^{0}(M(n), ω_{M(n)}^{∗}),

where, as usual, we denote MF (1)_{∗} simply by MF_{∗}.

5 Topological Modular Forms and Level Structures

By the obstruction theory of Goerss-Hopkins-Miller, as well as work of Lurie, the moduli stack M has been upgraded to a derived Deligne-Mumford stack, in such a way that the underlying ordinary geometry is respected. Namely, a proof of the following theorem can be found in Mark Behrens’ notes [Beh07].

Theorem 5.1 (Goerss-Hopkins-Miller, Lurie). The moduli stack M admits a sheaf of E_{∞}-rings O^{top} on its étale site which makes it into a derived Deligne-Mumford stack. For an étale map Spec R → M classifying a generalized elliptic curve C/R, the sections O^{top}(Spec R) form an even weakly periodic ring spectrum E such that

(a) π_{0}E = R, and

(b) the formal group G_{E} associated to E is isomorphic to the completion Ĉ at the identity section.

Moreover, there are isomorphisms of quasi-coherent π_{0}O^{top}-modules π_{2k}O^{top} ≅ ω_{M}^{k} and π_{2k+1}O^{top} ≅ 0 for all integers k.

The spectrum of topological modular forms T mf is defined to be the E_{∞}-ring spectrum of global sections RΓ(O^{top}).

We remark that inverting 6 kills all torsion in the cohomology of M as well as the homotopy of T mf. In that case, following the approach of this paper would fairly formally imply Anderson self-duality for T mf[1/6] from the fact that M has Grothendieck-Serre duality. To understand integral duality on the derived moduli stack of generalized elliptic curves, we would like to use the strategy of descent, dealing separately with the 2 and 3-torsion. The case when 2 is invertible captures the 3-torsion, is more tractable, and is thoroughly worked out in this paper by using the smallest good cover by level structures, M(2). The 2-torsion phenomena involve computations that are more daunting and will be dealt with subsequently. However, the same methodology works to give the required self-duality result. To begin, we need to lift M(2) and the covering map q : M(2) → M to the setting of derived Deligne-Mumford stacks. We point out that this is not immediate from Theorem 5.1 because the map q is not étale. However, we will explain how one can amend the construction to obtain a sheaf of E_{∞}- rings O(2)^{top} on M(2). We will also incorporate the GL_{2}(ℤ/2)-action, which is crucial for our result. We will in fact sketch an argument based on Mark Behrens’ [Beh07] to construct sheaves O(n)^{top} on M(n)[1/2n] for any n, as the extra generality does not complicate the solution. As we remarked earlier, the restriction of q to the smooth locus M(n)^{0} is an étale GL_{2}(ℤ/n)-torsor, hence we automatically obtain O(n)|_{M(n)^{0}} together with its GL_{2}(ℤ/n)-action. We will use the Tate curve and K(1)-local obstruction theory to construct the E_{∞}-ring of sections of the putative O(n)^{top} in a neighborhood of the cusps, and sketch a proof of the following theorem.

Theorem 5.2 (Goerss-Hopkins). The moduli stack M(n) (as an object over ℤ[1/2n]) admits a sheaf of even weakly periodic E_{∞}-rings O(n)^{top} on its étale site which makes it into a derived Deligne-Mumford stack. There are isomorphisms of quasi-coherent π_{0}O(n)^{top}-modules π_{2k}O(n)^{top} ≅ ω_{M(n)}^{k} and π_{2k−1}O(n)^{top} ≅ 0 for all integers k. Moreover, the covering map q : M(n) → M[1/2n] is a map of derived Deligne-Mumford stacks.

5.1 Equivariant K(1)-local Obstruction Theory

This is a combination of the Goerss-Hopkins’ [GH04b, GH04a] and Cooke’s [Coo78] obstruction theories, which in fact is contained although not explicitly stated in [GH04b, GH04a]. Let G be a finite group; a G-equivariant θ-algebra is an algebraic model for the p-adic K-theory of an E_{∞} ring spectrum with an action of G by E_{∞}-ring maps. Here, G-equivariance means that the action of G commutes with the θ-algebra operations. As G-objects are G-diagrams, the obstruction theory framework of [GH04a] applies. Let H_{G−θ}^{s}(A, B[t]) denote the s-th derived functor of derivations from A into B[t]^{4} in the category of G-equivariant θ-algebras. These are the G-equivariant derivatons from A into B[t], and there is a composite functor spectral sequence

4For a graded module B, B[t] is the shifted graded module with B[t]_{k} = B_{t+k}.

Documenta Mathematica 17 (2012) 271–311

H^{r}(G, H_{θ}^{s−r}(A, B[t])) ⇒ H_{G−θ}^{s}(A, B[t]).  (6)

Theorem 5.3 (Goerss-Hopkins). (a) Given a G − θ-algebra A, the obstructions to existence of a K(1)-local even-periodic E_{∞}-ring X with a Gaction by E_{∞}-ring maps, such that K_{∗}X ≅ A (as G − θ-algebras) lie in

H_{G−θ}^{s}(A, A[−s + 2]) for s ≥ 3.

The obstructions to uniqueness like in

H_{G−θ}^{s}(A, A[−s + 1]) for s ≥ 2.

(b) Given K(1)-local E_{∞}-ring G-spectra X and Y whose K-theory is pcomplete, and an equivariant map of θ-algebras f_{∗} : K_{∗}X → K_{∗}Y, the obstructions to lifting f_{∗} to an equivariant map of E_{∞}-ring spectra lie in

H_{G−θ}^{s}(K_{∗}X, K_{∗}Y [−s + 1]), for s ≥ 2,

while the obstructions to uniqueness of such a map lie in

H_{G−θ}^{s}(K_{∗}X, K_{∗}Y [−s]), for s ≥ 1.

(c) Given such a map f : X → Y, there exists a spectral sequence

H_{G−θ}^{s}(K_{∗}X, K_{∗}Y [t]) ⇒ π_{−t−s}(E_{∞}(X, Y)^{G}, f),

computing the homotopy groups of the space of equivariant E_{∞}-ring maps, based at f.

5.2 The Igusa Tower

Fix a prime p > 2 which does not divide n. Let M_{p} denote the p-completion of M, and let M^{∞} denote a formal neighborhood of the cusps of M_{p}. We will use the same embellishments for the moduli with level structures. The idea is to use the above Goerss-Hopkins obstruction theory to construct the E_{∞}-ring of sections over M(n)^{∞} of the desired O(n)^{top}, from the algebraic data provided by the Igusa tower, which will supply an equivariant θ-algebra. We will consider the moduli stack M(n) as an object over ℤ[1/n, ζ], for ζ a primitive n-th root of unity. The structure map M(n) → Spec ℤ[1/n, ζ] is given by the Weil pairing [KM85, 5.6]. The structure of M^{∞} as well as M(n)^{∞} is best understood by the Tate curves. We already mentioned that the singular locus is given by Néron n-gons; the n-Tate curve is a generalized elliptic curve in a neighborhood of the singular locus. It is defined over the ring ℤ[[q^{1/n}]], so that it is smooth when q is invertible and a Néron n-gon when q is zero. For details of the construction, the reader is referred to [DR73, VII].

From [DR73, VII] and [KM85, Ch 10], we learn that M^{∞} = Spf ℤ_{p}[[q]], and that M(n)^{∞} = ∐_{cusps} Spf ℤ_{p}[ζ][[q^{1/n}]]. The GL_{2}(ℤ/n)-action on M(n)^{∞} is un-

derstood by studying level structures on Tate curves, and is fully described in [KM85, Theorem 10.9.1]. The group U of upper-triangular matrices in

GL_{2}(ℤ/n) acts on B(n) = ℤ_{p}[ζ][[q^{1/n}]] by the roots of unity: ( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) ∈ U

sends q^{1/n} to ζ^{a}q^{1/n}. Note that then, the inclusion ℤ_{p}[[q]] → B(n) is Uequivariant, and M(n)^{∞} is represented by the induced GL_{2}(ℤ/n)-module A(n) = Hom_{U}(GL_{2}(ℤ/n), B(n)). Denote by M(p^{k}, n)^{∞} a formal neighborhood of the cusps in the moduli stack of generalized elliptic curves equipped with level structure maps

η : μ_{p^{k}} ⟶^{∼} C[p^{k}]

φ : (ℤ/n)^{2} ⟶^{∼} C[n].

Note that as we are working in a formal neighborhood of the singular locus, the curves C classified by M(p^{k}, n)^{∞} have ordinary reduction modulo p. As k runs through the positive integers, we obtain an inverse system called the Igusa tower, and we will write M(p^{∞}, n) for the inverse limit. It is the ℤ_{p}^{×}-torsor over M(n)^{∞} given by the formal affine scheme Spf Hom(ℤ_{p}^{×}, A(n)) [KM85, Theorem 12.7.1]. The Tate curve comes equipped with a canonical invariant differential, which makes ω_{M(n)^{∞}} isomorphic to the line bundle associated to the graded A(n)- module A(n)[1]. We define A(n)_{∗} to be the evenly graded A(n)-module, which in degree 2t is H^{0}(M(n)^{∞}, ω_{M(n)^{∞}}^{t}) ≅ A(n)[t]. Similarly we define A(p^{∞}, n)_{∗}. The modules A(p^{∞}, n)_{∗} = Hom(ℤ_{p}^{×}, A(n)_{∗}) have a natural GL_{2}(ℤ/n)- equivariant θ-algebra structure, with operations coming from the following maps

(ψ^{k})^{∗} : M(p^{∞}, n) → M(p^{∞}, n)
(C, η, φ) ↦ (C, η ∘ [k], φ),
(ψ^{p})^{∗} : M(p^{∞}, n) → M(p^{∞}, n)
(C, η, φ) ↦ (C^{(p)}, η^{(p)}, φ^{(p)}).

Here, C^{(p)} is the elliptic curve obtained from C by pulling back along the absolute Frobenius map on the base scheme. The multiplication by p isogeny [p] : C → C factors through C^{(p)} as the relative Frobenius F : C → C^{(p)} followed by the Verschiebung map V : C^{(p)} → C [KM85, 12.1]. The level structure φ^{(p)} : (ℤ/n)^{2} → C^{(p)} is the unique map making the diagram

\xymatrix{ & (ℤ/n)^{2} \ar[d]^{φ} \ar@{-->}[dl]_{φ^{(p)}} \\ C^{(p)} \ar[r]^{V} & C}

commute. Likewise, η^{(p)} is the unique level structure making an analogous diagram commute. More details on η^{(p)} are in [Beh07, Section 5]. The operation ψ^{k} comes from the action of ℤ_{p}^{×} on the induced module A(p^{∞}, n)_{∗} = Hom(ℤ_{p}^{×}, A(n)_{∗}). We point out that from this description of the operations, it is clear that the GL_{2}(ℤ/n)-action commutes with the operations ψ^{k} and ψ^{p}, which in particular gives an isomorphism

A(p^{∞}, n)_{∗} ≅ Hom_{U} (G, Hom (ℤ_{p}^{×}, B(n)_{∗})) .  (7)

Denote by B(p^{∞}, n)_{∗} the θ-algebra Hom(ℤ_{p}^{×}, B(n)_{∗}). As a first step, we apply (a) of Theorem 5.3 to construct even-periodic K(1)- local GL_{2}(ℤ/n)-equivariant E_{∞}-ring spectra T mf(n)_{p}^{∞} whose p-adic K-theory is given by A(p^{∞}, n)_{∗}. The starting point is the input to the spectral sequence (6), the group cohomology

H^{r} (GL_{2}(ℤ/n), H_{θ}^{s−r} (A(p^{∞}, n)_{∗}, A(p^{∞}, n)_{∗})) .  (8)

Remark 5.4. Lemma 7.5 of [Beh07] implies that H_{θ}^{s}(B(p^{∞}, n)_{∗}, B(p^{∞}, n)_{∗}) = 0 for s > 1, from which we deduce the existence of a unique K(1)-local weakly even periodic E_{∞}-ring spectrum that we will denote by K[[q^{1/n}]], whose Ktheory is the θ-algebra B(p^{∞}, n)_{∗}. This spectrum K[[q^{1/n}]] should be thought of as the sections of O(n)^{top} over a formal neighborhood of a single cusp of M(n).

By the same token, we also know that H_{θ}^{s}(A(p^{∞}, n)_{∗}, A(p^{∞}, n)_{∗}) = 0 for s > 1, and

H_{θ}^{0} (A(p^{∞}, n)_{∗}, A(p^{∞}, n)_{∗}) ≅ Hom_{U} (GL_{2}(ℤ/n), H_{θ}^{0} (A(p^{∞}, n)_{∗}, B(p^{∞}, n)_{∗})) .

Thus the group cohomology (8) is simply

H^{r} (U, H_{θ}^{0} (A(p^{∞}, n)_{∗}, B(p^{∞}, n)_{∗}))

which is trivial for r > 0 as the coefficients are p-complete, and the group U has order n, coprime to p. Therefore, the spectral sequence (6) collapses to give that

H_{GL_{2}(ℤ/n)−θ}^{s} (A(p^{∞}, n)_{∗}, A(p^{∞}, n)_{∗}) = 0, for s > 0.

Applying Theorem 5.3 now gives our required GL_{2}(ℤ/n)-spectra T mf(n)_{p}^{∞}. A similar argument produces a GL_{2}(ℤ/n)-equivariant E_{∞}-ring map

q^{∞} : T mf_{p}^{∞} → T mf(n)_{p}^{∞},

where T mf^{∞} is given the trivial GL_{2}(ℤ/n)-action.

Proposition 5.5. The map q^{∞} : T mf_{p}^{∞} → T mf(n)_{p}^{∞} is the inclusion of homotopy fixed points.

Proof. Note that from our construction it follows that T mf(n)_{p}^{∞} is equivalent to Hom_{U}(GL_{2}(ℤ/n), K[[q^{1/n}]]), where K[[q^{1/n}]] has the U-action lifting the one we described above on its θ-algebra B(n)_{∗}. (This action lifts by obstruction theory to the E_{∞}-level because the order of U is coprime to p.) Since T mf_{p}^{∞} has trivial GL_{2}(ℤ/n)-action, the map q^{∞} factors through the homotopy fixed point spectrum

(T mf(n)_{p}^{∞})^{hGL_{2}(ℤ/n)} ≅ K[[q^{1/n}]]^{hU}.

So we need that the map q′ : T mf^{∞} = K[[q]] → K[[q^{1/n}]]^{hU} be an equivalence. The homotopy groups of K[[q^{1/n}]]^{hU} are simply the U-invariant homotopy in K[[q^{1/n}]], because U has no higher cohomology with p-adic coefficients. Thus π_{∗}q′ is an isomorphism, and the result follows.

5.3 Gluing

We need to patch these results together to obtain the sheaf O(n)^{top} on the étale site of M(n). To construct the presheaves Õ(n)_{p}^{top} on the site of affine schemes étale over M(n), one follows the procedure of [Beh07, Step 2, Sections 7 and 8]. Thus for each prime p > 2 which does not divide n, we have Õ(n)_{p} on M(n)_{p}. As in [Beh07, Section 9], rational homotopy theory produces a presheaf Õ(n)_{ℚ}. These glue together to give Õ(n) (note, 2n will be invertible in Õ(n)). By construction, the homotopy group sheaves of Õ(n)^{top} are given by the tensor powers of the sheaf of invariant differentials ω_{M(n)}, which is a quasi-coherent sheaf on M(n). Section 2 of [Beh07] explains how this data gives rise to a sheaf O(n)^{top} on the étale site of M(n), so that the E_{∞}-ring spectrum T mf(n) of global sections of O(n)^{top} can also be described as follows. Denote by T MF (n) the E_{∞}-ring spectrum of sections RΓ (O(n)^{top}|_{M(n)^{0}}) over the locus of smooth curves. Let T MF (n)^{∞} be the the E_{∞}-ring of sections of O(n)^{top}|_{M(n)^{0}} in a formal neighborhood of the cusps. Both have a GL_{2}(ℤ/n)-action by construction. The equivariant obstruction theory of Theorem 6 can be used again to construct a GL_{2}(ℤ/n)-equivariant map T MF (n)_{K(1)} → T MF (n)_{p}^{∞} that refines the q-expansion map; again, the key point as that K_{∗}(T MF (n)_{p}^{∞}) is a U-induced GL_{2}(ℤ/n)-module. Precomposing with the K(1)-localization map, we get a GL_{2}(ℤ/n)-E_{∞}-ring map T MF (n) → T MF (n)^{∞}. We build T mf(n)_{p} as a pullback

\xymatrix{T mf(n)_{p} \ar[r] \ar[d] & T mf(n)_{p}^{∞} \ar[d] \\ T MF (n)_{p} \ar[r] & T MF (n)_{p}^{∞}}

All maps involved are GL_{2}(ℤ/n)-equivariant maps of E_{∞}-rings, so T mf(n) constructed this way has a GL_{2}(ℤ/n)-action as well.

6 Descent and Homotopy Fixed Points

We have remarked several times that the map q^{0} : M(n)^{0} → M^{0} is a GL_{2}(ℤ/n)-torsor, thus we have a particularly nice form of étale descent. On global sections, this statement translates to the equivalence

T MF [1/2n] → T MF (n)^{hGL_{2}(ℤ/n)}.

The remarkable fact is that this property goes through for the compactified version as well.

Theorem 6.1. The map T mf[1/2n] → T mf(n)^{hGL_{2}(ℤ/n)} is an equivalence.

Proof. This is true away from the cusps, but by Proposition 5.5, it is also true near the cusps. We constructed T mf(n) from these two via pullback diagrams, and homotopy fixed points commute with pullbacks.

Remark 6.2. For the rest of this paper, we will investigate T mf(2), the spectrum of topological modular forms with level 2 structure. Note that this spectrum differs from the more commonly encountered T MF_{0}(2), which is the receptacle for the Ochanine genus [Och87], as well as the spectrum appearing in the resolution of the K(2)-local sphere [GHMR05, Beh06]. The latter is obtained by considering isogenies of degree 2 on elliptic curves, so-called Γ_{0}(2) structures.

7 Level 2 Structures Made Explicit

In this section we find an explicit presentation of the moduli stack M(2). Let E/S be a generalized elliptic curve over a scheme on which 2 is invertible, and whose geometric fibers are either smooth or have a nodal singularity (i.e. are Néron 1-gons). Then Zariski locally, E is isomorphic to a Weierstrass curve of a specific and particularly simple form. Explicitly, there is a cover U → S and functions x, y on U such that the map U → ℙ_{U}^{2} given by [x, y, 1] is an isomorphism between E_{U} = E ×_{S} U and a Weierstrass curve in ℙ_{U}^{2} given by the

equation

E_{b} : y^{2} = x^{3} + \frac{b_{2}}{4} x^{2} + \frac{b_{4}}{2} x + \frac{b_{6}}{4} =: f_{b}(x),  (9)

such that the identity for the group structure on E_{U} is mapped to the point at infinity [0, 1, 0][Sil86, III.3],[KM85]. Any two Weierstrass equations for E_{U} are related by a linear change of variables of the form

\begin{matrix}x & ↦ u^{−2}x + r \\ y & ↦ u^{−3}y. \end{matrix}  (10)

The object which classifies locally Weierstrass curves of the form (9), together with isomorphisms which are given as linear change of variables (10), is a stack M_{weier}[1/2], and the above assignment E ↦ E_{b} of a locally Weierstrass curve to an elliptic curve defines a map w : M[1/2] → M_{weier}[1/2]. The Weierstrass curve (9) associated to a generalized elliptic curve E over an algebraically closed field has the following properties: E is smooth if and only if the discriminant of f_{b}(x) has no repeated roots, and E has a nodal singularity if and only if f_{b}(x) has a repeated but not a triple root. Moreover, non-isomorphic elliptic curves cannot have isomorphic Weierstrass presentations. Thus the map w : M[1/2] → M_{weier}[1/2] injects M[1/2] into the open substack U(Δ) of M_{weier}[1/2] which is the locus where the discriminant of f_{b} has order of vanishing at most one. Conversely, any Weierstrass curve of the form (9) has genus one, is smooth if and only if f_{b}(x) has no repeated roots, and has a nodal singularity whenever it has a double root, so w : M[1/2] → U(Δ) is also surjective, hence an isomorphism. Using this and the fact that points of order two on an elliptic curve are well understood, we will find a fairly simple presentation of M(2). The moduli stack of locally Weierstrass curves is represented by the Hopf algebroid

(B = ℤ[1/2][b_{2}, b_{4}, b_{6}], B[u^{±1}, r]).

Explicitly, there is a presentation Spec B → M_{weier}[1/2], such that

Spec B ×_{M_{weier}[1/2]} Spec B = Spec B[u^{±1}, r].

The projection maps to Spec B are Spec of the inclusion of B in B[u^{±1}] and Spec of the map

b_{2} ↦ u^{2}(b_{2} + 12r)
b_{4} ↦ u^{4}(b_{4} + rb_{2} + 6r^{2})
b_{6} ↦ u^{6}(b_{6} + 2rb_{4} + r^{2}b_{2} + 4r^{3})

which is obtained by plugging in the transformation (10) into (9). In other words, M_{weier}[1/2] is simply obtained from Spec B by enforcing the isomorphisms that come from the change of variables (10). Suppose E/S is a smooth elliptic curve which is given locally as a Weierstrass curve (9), and let ϕ : (ℤ/2)^{2} → E be a level 2 structure. For convenience in the notation, define e_{0} = \binom{1}{1}, e_{1} = \binom{1}{0}, e_{2} = \binom{0}{1} ∈ (ℤ/2)^{2}. Then ϕ(e_{i}) are all points of exact order 2 on E, thus have y-coordinate equal to zero since [−1](y, x) = (−y, x) ([Sil86, III.2]) and (9) becomes

y^{2} = (x − x_{0})(x − x_{1})(x − x_{2}),  (11)

where x_{i} = x(ϕ(e_{i})) are all different. If E is a generalized elliptic curve which is singular, i.e. E is a Néron 2-gon, then a choice of level 2 structure makes E locally isomorphic to the blow-up of (11) at the singularity (seen as a point of ℙ^{2}), with x_{i} = x_{j} ≠ x_{k}, for {i, j, k} = {0, 1, 2}. So let A = ℤ[1/2][x_{0}, x_{1}, x_{2}], let L be the line in Spec A defined by the ideal (x_{0} − x_{1}, x_{1} − x_{2}, x_{2} − x_{0}), and let Spec A − L be the open complement. The change of variables (10) translates to a (𝔾_{a} ⋊ 𝔾_{m})-action on Spec A that preserves L and is given by:

x_{i} ↦ u^{2}(x_{i} − r).

Consider the isomorphism ψ : (Spec A − L) ⟶^{∼} (𝔸^{2} − {0}) × 𝔸^{1}:

(x_{0}, x_{1}, x_{2}) ↦ ((x_{1} − x_{0}, x_{2} − x_{0}), x_{0}) .

We see that 𝔾_{a} acts trivially on the (𝔸^{2} −{0})-factor, and freely by translation on 𝔸^{1}. Therefore the quotient (Spec A − L)//𝔾_{a} is

M̃(2) = 𝔸^{2} − {0} = Spec ℤ[1/2][λ_{1}, λ_{2}] − {0},

the quotient map being ψ composed with the projection onto the first factor. This corresponds to choosing coordinates in which E is of the form:

y^{2} = x(x − λ_{1})(x − λ_{2}).  (12)

The 𝔾_{m}-action is given by grading A as well as Λ = ℤ[1/2][λ_{1}, λ_{2}] so that the degree of each x_{i} and λ_{i} is 2. It follows that M(2) = M̃(2)//𝔾_{m} is the weighted projective line Proj Λ = (Spec Λ−{0})//𝔾_{m}. Note that we are taking homotopy quotient which makes a difference: −1 is a non-trivial automorphism on M(2) of order 2. The sheaf ω_{M(2)} is an ample invertible line bundle on M(2), locally gen-

erated by the invariant differential η_{E_{λ}} = \frac{dx}{2y}. From (10) we see that the

𝔾_{m} = Spec ℤ[u^{±1}]-action changes η_{E_{λ}} to uη_{E_{λ}}. Hence, ω_{M(2)} is the line bundle on M(2) = Proj Λ which corresponds to the shifted module Λ[1], standardly denoted by O(1). We summarize the result.

Proposition 7.1. The moduli stack of generalized elliptic curves with a choice of a level 2 structure M(2) is isomorphic to Proj Λ = (Spec Λ − {0})//𝔾_{m}, via the map M(2) → Proj Λ which classifies the sheaf of invariant differentials ω_{M(2)}on M(2). The universal curve over the locus of smooth curves M(2)^{0} = Proj Λ − {0, 1, ∞} is the curve of equation (12). The fibers at 0, 1, and ∞, are Néron 2-gons obtained by blowing up the singularity of the curve (12).

Remark 7.2. As specifying a 𝔾_{m}-action is the same as specifying a grading, we can think of the ringed space (M(2), ω_{M(2)}^{∗}) as the ringed space (M̃(2) = Spec Λ − {0}, O_{M̃(2)}) together with the induced grading.

Next we proceed to understand the action of GL_{2}(ℤ/2) on the global sections H^{0}(M(2), ω_{M(2)}^{∗}) = Λ. By definition, the action comes from the natural action of GL_{2}(ℤ/2) on (ℤ/2)^{2} and hence by pre-composition on the level structure maps ϕ : (ℤ/2)^{2} → E[2]. If we think of GL_{2}(ℤ/2) as the symmetric group S_{3}, then this action permutes the non-zero elements {e_{0}, e_{1}, e_{2}} of (ℤ/2)^{2}, which translates to the action on

H^{0}(Spec A − L, O_{Spec A−L}) = ℤ[x_{0}, x_{1}, x_{2}],

given as g · x_{i} = x_{gi} where g ∈ S_{3} = Perm{0, 1, 2}. The map on H^{0} induced by the projection (Spec A − L) → M̃(2) is

ℤ[λ_{1}, λ_{2}] → ℤ[x_{0}, x_{1}, x_{2}]
λ_{i} ↦ x_{i} − x_{0}

Therefore, we obtain that gλ_{i} is the inverse image of x_{gi} − x_{g0}. That is, gλ_{i} = λ_{gi} − λ_{g0}, where we implicitly understand that λ_{0} = 0. We have proven the following lemma.

Lemma 7.3. Choose the generators of S_{3} = Perm{0, 1, 2}, σ = (012) and τ = (12). Then the S_{3} action on Λ = H^{0}(M(2), ω_{M(2)}^{∗}) is determined by

\begin{aligned}τ : & λ_{1} ↦ λ_{2} & σ : & λ_{1} ↦ λ_{2} − λ_{1} \\ & λ_{2} ↦ λ_{1} & & λ_{2} ↦ −λ_{1}. \end{aligned}

This fully describes the global sections H^{0}(M(2), ω_{M(2)}^{∗}) as an S_{3}-module. The action on H^{1}(M(2), ω_{M(2)}^{∗}) is not as apparent and we deal with it using Serre duality (8.5).

8 (Equivariant) Serre Duality for M(2)

We will proceed to prove Serre duality for M(2) in an explicit manner that will be useful later, by following the standard computations for projective spaces, as in [Har77]. To emphasize the analogy with the corresponding statements about the usual projective line, we might write Proj Λ and O(∗) instead of M(2) and ω_{M(2)}^{∗}, in view of Remark (7.2). Also remember that for brevity, we might omit writing 1/2.

Proposition 8.1. The cohomology of M(2) with coefficients in the graded sheaf of invariant differentials ω_{M(2)}^{∗} is computed as

H^{s}(M(2), ω_{M(2)}^{∗}) = \begin{cases}Λ, & s = 0 \\ Λ/(λ_{1}^{∞}, λ_{2}^{∞}), & s = 1 \\ 0, & else \end{cases}

where Λ/(λ_{1}^{∞}, λ_{2}^{∞}) is a torsion Λ-module with a ℤ[1/2]-basis of monomials _{\frac{1}{λ_{1}^{i}λ_{2}^{j}}} for i, j both positive.

Remark 8.2. The module Λ/(λ_{1}^{∞}, λ_{2}^{∞}) is inductively defined by the short exact sequences

0 →Λ ⟶^{λ_{1}} Λ[\frac{1}{λ_{1}} ] → Λ/(λ_{1}^{∞}) → 0
0 →Λ/(λ_{1}^{∞}) ⟶^{λ_{2}} Λ[\frac{1}{λ_{1}λ_{2}} ] → Λ/(λ_{1}^{∞}, λ_{2}^{∞}) → 0.

Remark 8.3. Note that according to 7.2, H^{∗}(M(2), ω_{M(2)}^{∗}) is isomorphic to H^{∗}(M̃(2), O_{M̃(2)}) with the induced grading. It is these latter cohomology groups that we shall compute.

Proof. We proceed using the local cohomology long exact sequence [Har77, Ch III, ex. 2.3] for

M̃(2) ⊂ Spec Λ ⊃ {0}.

The local cohomology groups R^{∗}Γ_{{0}}(Spec Λ, O) are computed via a Koszul complex as follows. The ideal of definition for the point {0} ∈ Spec Λ is (λ_{1}, λ_{2}), and the generators λ_{i} form a regular sequence. Hence, R^{∗}Γ_{{0}}(Spec Λ, O) is the cohomology of the Koszul complex

Λ → Λ[\frac{1}{λ_{1}} ] × Λ[\frac{1}{λ_{2}} ] → Λ[\frac{1}{λ_{1}λ_{2}} ],

which is Λ/(λ_{1}^{∞}, λ_{2}^{∞}), concentrated in (cohomological) degree two. We also know that H^{∗}(Spec Λ, O) = Λ concentrated in degree zero, so that the local cohomology long exact sequence splits into

0 → Λ → H^{0}(M̃(2), O) → 0
0 → H^{1}(M̃(2), O) → Λ/(λ_{1}^{∞}, λ_{2}^{∞}) → 0,

giving the result.

Lemma 8.4. We have the following properties of the sheaf of differentials on M(2).

(a) There is an isomorphism Ω_{M(2)} ≅ ω_{M(2)}^{−4}.

(b) The cohomology group H^{s}(M(2), Ω_{M(2)}) is zero unless s = 1, and H^{1}(M(2), Ω_{M(2)}) is the sign representation ℤ_{sgn}[1/2] of S_{3}.

Proof. (a) The differential form η = λ_{1}dλ_{2} − λ_{2}dλ_{1} is a nowhere vanishing differential form of degree four, thus a trivializing global section of the sheaf O(4) ⊗ Ω_{Proj Λ}. Hence there is an isomorphism Ω_{Proj Λ} ≅ O(−4).

(b) From Proposition 8.1, H^{∗}(Proj Λ, Ω) is ℤ[1/2] concentrated in cohomological degree one, and generated by

\frac{η}{λ_{1}λ_{2}} = \frac{λ_{1}}{λ_{2}} d(\frac{λ_{2}}{λ_{1}} ).

Any projective transformation φ of Proj Λ acts on H^{1}(Proj Λ, Ω) by the determinant det φ. By our previous computations, as summarized in Lemma 7.3, the transpositions of S_{3} act with determinant −1, and the elements of order 3 of S_{3} with determinant 1. Hence the claim.

We are now ready to state and prove the following result.

Theorem 8.5 (Serre Duality). The sheaf of differentials Ω_{M(2)} is a dualizing sheaf on M(2), i.e. the natural cup product map

H^{0}(M(2), ω_{M(2)}^{t}) ⊗ H^{1}(M(2), ω_{M(2)}^{−t} ⊗ Ω_{M(2)}) → H^{1}(M(2), Ω_{M(2)}),

is a perfect pairing which is compatible with the S_{3}-action.

Remark 8.6. Compatibility with the S_{3}-action simply means that for every g ∈ S_{3}, the following diagram commutes

\xymatrix{H^{0}(M(2), g^{∗}ω_{M(2)}^{t}) ⊗ H^{1}(M(2), g^{∗}ω_{M(2)}^{−t} ⊗ g^{∗}Ω_{M(2)}) \ar[r] \ar[d]_{g} & H^{1}(M(2), g^{∗}Ω_{M(2)}) \ar[d]^{g} \\ H^{0}(M(2), ω_{M(2)}^{t}) ⊗ H^{1}(M(2), ω_{M(2)}^{−t} ⊗ Ω_{M(2)}) \ar[r] & H^{1}(M(2), Ω_{M(2)})} .

But we have made a choice of generators for Λ = H^{0}(M(2), ω_{M(2)}^{∗}) ≅ H^{0}(M(2), g^{∗}ω_{M(2)}^{∗}), and we have described the S_{3}-action on those generators in Lemma 7.3. If we think of the induced maps g : H^{∗}(M(2), g^{∗}ω_{M(2)}^{∗}) → H^{∗}(M(2), ω_{M(2)}^{∗}) as a change of basis action of S_{3}, Theorem 8.5 states that we have a perfect pairing of S_{3}-modules. As a consequence, there is an S_{3}-module isomorphism

H^{1}(M(2), ω_{M(2)}^{∗−4}) ≅ Hom(Λ, ℤ_{sgn}[1/2]) = Λ_{sgn}^{∨}.

(The subscript sgn will always denote twisting by the sign representation of S_{3}.)

Proof. Proposition 8.1 and Lemma 8.4 give us explicitly all of the modules involved. Namely, H^{0}(Proj Λ, O(∗)) is free on the monomials λ_{1}^{i}λ_{2}^{j}, for i, j ≥ 0, and H^{1}(Proj Λ, O(∗)) is free on the monomials _{\frac{1}{λ_{1}^{i}λ_{2}^{j}}} = _{\frac{1}{λ_{1}^{i−1}λ_{2}^{j−1}}} \frac{1}{λ_{1}λ_{2}} , for i, j >

0. Lemma (8.4) gives us in addition that H^{1}(Proj Λ, O(∗) ⊗ Ω_{M(2)}) is free on \frac{1}{λ_{1}^{i−1}λ_{2}^{j−1}} \frac{η}{λ_{1}λ_{2}} . We conclude that

(λ_{1}^{i}λ_{2}^{j}, \frac{η}{λ_{1}^{i+1}λ_{2}^{j+1}} ) ↦ \frac{η}{λ_{1}λ_{2}}

is really a perfect pairing. Moreover, this pairing is compatible with any projective transformation φ of Proj Λ, which includes the S_{3}-action as well as change of basis. Any such φ acts on H^{∗}(Proj Λ, O(∗)) by a linear change of variables, and changes η by the determinant det φ. Thus the diagram

\xymatrix{H^{0}(M(2), φ^{∗}ω_{M(2)}^{t}) ⊗ H^{1}(M(2), φ^{∗}ω_{M(2)}^{−t} ⊗ φ^{∗}Ω_{M(2)}) \ar[r] \ar[d]_{φ} \ar[d]^{φ⊗det φ} & H^{1}(M(2), φ^{∗}Ω_{M(2)}) \ar[d]^{det φ} \\ H^{0}(M(2), ω_{M(2)}^{t}) ⊗ H^{1}(M(2), ω_{M(2)}^{−t} ⊗ Ω_{M(2)}) \ar[r] & H^{1}(M(2), Ω_{M(2)})}

commutes.

We explicitly described the induced action on the global sections H^{0}(Proj Λ, O(∗)) = Λ in Lemma 7.3, and in (8.4) we have identified H^{1}(Proj Λ, Ω_{Proj Λ}) with the sign representation ℤ_{sgn} of S_{3}. Therefore, the perfect pairing is the natural map

Λ ⊗ Λ_{sgn}^{∨} → ℤ_{sgn}[1/2].

9 Anderson Duality for T mf(2)

The above Serre duality pairing for M(2) enables us to compute the homotopy groups of T mf(2) as a module over S_{3}. We obtain that the E_{2} term of the spectral sequence (5) for T mf(2) looks as follows:

Figure 1: Jardine spectral sequence (5) for π_{∗}T mf(2)

As there is no space for differentials or extensions, we conclude that

π_{∗}T mf(2) = Λ ⊕ Σ^{−9}Λ_{sgn}^{∨}.

Even better, we are now able to prove self-duality for T mf(2). Since we are working with 2 inverted everywhere, the Anderson dual of T mf(2) is defined by dualizing the homotopy groups as ℤ[1/2]-modules, as noted in Remark 2.1. Although this duality may deserve the notation I_{ℤ[1/2]}, we forbear in the interest of compactness of the notation. Recall that the Anderson dual of T mf(2) is the function spectrum F (T mf(2), I_{ℤ}), so it inherits an action by S_{3} from the one on T mf(2).

Theorem 9.1. The Anderson dual of T mf(2) is Σ^{9}T mf(2). The inherited S_{3}-action on π_{∗}I_{ℤ}T mf(2) corresponds to the action on π_{∗}Σ^{9}T mf(2) up to a twist by sign.

Proof. On the level of homotopy groups, we have the spectral sequence (2) which collapses because each π_{i}T mf(2) is a free (and dualizable) ℤ-module. Thus

π_{∗}I_{ℤ}T mf(2) = (Λ ⊕ Σ^{−9}Λ_{sgn}^{∨})^{∨},

which is isomorphic to Σ^{9}(π_{∗}T mf(2))_{sgn} = Λ^{∨}⊕Σ^{9}Λ_{sgn}, as π_{∗}T mf(2)-modules via a double duality map. Now, I_{ℤ}T mf(2), being defined as the function spectrum F (T mf(2), I_{ℤ}), is naturally a T mf(2)-module, thus a dualizing class f : S^{9} → I_{ℤ}T mf(2) extends to an equivalence f̃ : Σ^{9}T mf(2) → I_{ℤ}T mf(2). Specifically, let f : S^{9} → I_{ℤ}T mf(2) represent a generator of π_{9}I_{ℤ}T mf(2) = ℤ[1/2], which is also a generator of π_{∗}I_{ℤ}T mf(2) as a π_{∗}T mf(2)-module. Then the composition

f̃: S^{9} ∧ T mf(2) ⟶^{f∧Id_{T mf(2)}} I_{ℤ}T mf(2) ∧ T mf(2) ⟶^{ψ} I_{ℤ}T mf(2),

where ψ is the T mf(2)-action map, gives the required equivalence. Namely, let a be an element of π_{∗}T mf(2); then f̃_{∗}(Σ^{9}a) = [f]a, but f was chosen so that its homotopy class generates π_{∗}I_{ℤ}T mf(2) as a π_{∗}T mf(2)-module.

10 Group Cohomology Computations

This section is purely technical; for further use, we compute the S_{3} homology and cohomology of the module H^{∗}(M(2), ω_{M(2)}^{∗}) = Λ ⊕ Σ^{−9}Λ_{sgn}^{∨}, where the action is described in Lemma 7.3. First we deal with Tate cohomology, and then we proceed to compute the invariants and coinvariants.

10.1 Tate Cohomology

The symmetric group on three letters fits in a short exact sequence

1 → C_{3} → S_{3} → C_{2} → 1,

producing a Lyndon-Hochschield-Serre spectral sequence for the cohomology of S_{3}. If 2 is invertible in the S_{3}-module M, the spectral sequence collapses to give that the S_{3} cohomology, as well as the S_{3}-Tate cohomology, is computed as the fixed points of the C_{3}-analogue

H^{∗}(C_{3}, M)^{C_{2}} ≅ H^{∗}(S_{3}, M)
Ĥ^{∗}(C_{3}, M)^{C_{2}} ≅ Ĥ^{∗}(S_{3}, M).

Therefore, it suffices to compute the respective C_{3}-cohomology groups as C_{2}- modules. To do this, we proceed as in [GHMR05]. Give A = ℤ[1/2][x_{0}, x_{1}, x_{2}] the left S_{3}-action as follows: g ∈ S_{3} maps x_{i} to (−1)^{sgn g}x_{gi}. We have a surjection of S_{3}-modules A → Λ given by

x_{0} ↦ λ_{1}
σ(x_{0}) = x_{1} ↦ λ_{2} − λ_{1} = σ(λ_{1})
σ^{2}(x_{0}) = x_{2} ↦ −λ_{2} = σ^{2}(λ_{2}).

The kernel of this map is the ideal generated by σ_{1} = x_{0} + x_{1} + x_{2}. Therefore, we have a short exact sequence

0 → Aσ_{1} → A → Λ → 0.  (13)

The orbit under σ of each monomial of A has 3 elements, unless that polynomial is a power of σ_{3} = x_{0}x_{1}x_{2}. Therefore, A splits as a sum of a S_{3}-module F with free C_{3}-action and ℤ[σ_{3}] which has trivial C_{3}-action, i.e.

A = F ⊕ ℤ[σ_{3}].  (14)

Let N : A → H^{0}(C_{3}, A) be the additive norm map, and let d denote the cohomology class in bidegree (0, 6) represented by σ_{3}. Then we have an exact sequence

A ⟶^{N} H^{∗}(C_{3}, A) → ℤ/3[b, d] → 0,

where b is a cohomology class of bidegree (2, 0). The Tate cohomology of A is then

Ĥ^{∗}(C_{3}, A) ≅ H^{∗}(C_{3}, A)[b^{−1}] →^{∼} ℤ/3[b^{±1}, d].

The quotient C_{2}-action is given by τ(b) = −b and τ(d) = −d. Similarly, noting that the degree of σ_{1} is 2, and τ(σ_{1}) = −σ_{1}, we obtain that the C_{3}-cohomology of the module Aσ_{1} is the same as that of A, with the internal grading shifted by 2, and the quotient C_{2}-action twisted by sign. In other words,

Ĥ^{∗}(C_{3}, Aσ_{1}) ≅ Σ^{2} ((ℤ_{sgn}/3)[b̃^{±1}, d̃]) .

where again b̃ and d̃ have bidegrees (2, 0) and (0, 6) respectively, and the quotient C_{2}-action is described by

τ : b̃^{i}d̃^{j} ↦ (−1)^{i+j+1}b̃^{i}d̃^{j}.

Note that Ĥ^{∗}(C_{3}, A) and Ĥ^{∗}(C_{3}, Aσ_{1}) are concentrated in even cohomological degrees. Therefore, the long exact sequence in cohomology induced by (13) breaks up into the exact sequences

0 → Ĥ^{2k−1}(C_{3}, Λ) → Ĥ^{2k}(C_{3}, Aσ_{1}) → Ĥ^{2k}(C_{3}, A) → Ĥ^{2k}(C_{3}, Λ) → 0. (15)

The middle map in this exact sequence is zero, because it is induced by multiplication by σ_{1}, which is in the image of the additive norm on A. It follows that

Ĥ^{∗}(C_{3}, Λ) ≅ ℤ/3[a, b^{±1}, d]/(a^{2}),

where a is the element in bidegree (1, 2) which maps to b̃ ∈ Ĥ^{2}(C_{3}, Aσ_{1}). The quotient action by C_{2} is described as

τ : a ↦ a
b ↦ −b
d ↦ −d.

Now it only remains to take fixed points to compute the Tate cohomology of Λ and Λ_{sgn}.

Proposition 10.1. Denote by R the graded ring ℤ/3[a, b^{±2}, d^{2}]/(a^{2}). Then Tate cohomology of the S_{3}-modules Λ and Λ_{sgn} is

Ĥ^{∗}(S_{3}, Λ) = R ⊕ Rbd,
Ĥ^{∗}(S_{3}, Λ_{sgn}) = Rb ⊕ Rd.

Remark 10.2. The classes a and bd will represent the elements of π_{∗}T mf commonly known as α and β, respectively, at least up to a unit.

10.2 Invariants

We now proceed to compute the invariants H^{0}(S_{3}, H^{∗}(M(2), ω_{M(2)}^{∗})). The result is summarized in the next proposition.

Proposition 10.3. The invariants of Λ under the S_{3}-action are isomorphic to the ring of modular forms MF_{∗}[1/2], i.e.

Λ^{S_{3}} = ℤ[1/2][c_{4}, c_{6}, Δ]/(1728Δ − c_{4}^{3} − c_{6}^{2}).

The twisted invariants module Λ_{sgn}^{S_{3}} is a free Λ^{S_{3}}-module on a generator d of degree 6.

Proof. Let ε ∈ A denote the alternating polynomial (x_{1} −x_{2})(x_{1} −x_{3})(x_{2} −x_{3}). Then ε^{2} is symmetric, so it must be a polynomial g(σ_{1}, σ_{2}, σ_{3}) in the elementary symmetric polynomials. Indeed, g is the discriminant of the polynomial

(x − x_{0})(x − x_{1})(x − x_{2}) = x^{3} + σ_{1}x^{2} + σ_{2}x + σ_{3}. (11)

The C_{3} invariants in A are the alternating polynomials in three variables

A^{C_{3}} = ℤ[σ_{1}, σ_{2}, σ_{3}, ε]/(ε^{2} − g).

The quotient action by C_{2} fixes σ_{2} and ε, and changes the sign of σ_{1} and σ_{3}. Since C_{3} fixes σ_{1}, the invariants in the ideal in A generated by σ_{1} are the ideal generated by σ_{1} in the invariants A^{C_{3}}. As H^{1}(C_{3}, Aσ_{1}) = 0, the long exact sequence in cohomology gives a short exact sequence of invariants

σ_{1}A^{C_{3}} → A^{C_{3}} → Λ^{C_{3}} → 0.

Denoting by p the quotient map A^{C_{3}} → Λ^{C_{3}}, we now have

Λ^{C_{3}} ≅ A^{C_{3}}/(σ_{1}) = ℤ[p(σ_{2}), p(σ_{3}), p(ε)]/p(ε^{2} + 27σ_{3}^{2} + 4σ_{2}^{3})

where τ fixes p(σ_{2}) and p(ε) and changes the sign of p(σ_{3}). It is consistent with the above computations of Tate cohomology to denote σ_{3} and p(σ_{3}) by d. The invariant quantities are well-known; they are the modular forms of E_{λ} of (12), the universal elliptic curve over M(2):

\begin{matrix}p(σ_{2}) = & − (λ_{1}^{2} + λ_{2}^{2} − λ_{1}λ_{2}) = −\frac{1}{16}c_{4} \\ p(ε) = & − (λ_{1} + λ_{2})(2λ_{1}^{2} + 2λ_{2}^{2} − 5λ_{1}λ_{2}) = \frac{1}{32}c_{6} \\ p(σ_{3}^{2}) = & d^{2} = λ_{1}^{2}λ_{2}^{2}(λ_{2} − λ_{1})^{2} = \frac{1}{16}Δ. \end{matrix}  (16)

Hence, d is a square root of the discriminant Δ, and since 2 is invertible, we get that the invariants

\begin{aligned}Λ^{S_{3}} & = ℤ[1/2][p(σ_{2}), p(σ_{3}^{2}), p(ε)]/p(ε^{2} + 27σ_{3}^{2} + 4σ_{2}^{3}) \\ & = ℤ[1/2][c_{4}, c_{6}, Δ]/(1728Δ − c_{4}^{3} + c_{6}^{2}) = MF_{∗} \end{aligned}  (17)

are the ring of modular forms, as expected. Moreover, there is a splitting Λ^{C_{3}} ≅ Λ^{S_{3}} ⊕ dΛ^{S_{3}}, giving that

Λ_{sgn}^{S_{3}} = dΛ^{S_{3}}.  (18)

10.3 Coinvariants and Dual Invariants

To be able to use Theorem (8.5) to compute homotopy groups, we also need to know the S_{3}-cohomology of the signed dual of Λ. For this, we can use the composite functor spectral sequence for the functors Hom_{ℤ}(−, ℤ) and ℤ ⊗_{ℤS_{3}} (−).

Since Λ is free over ℤ, we get that

Hom_{ℤ}(ℤ ⊗_{ℤS_{3}} Λ_{sgn}, ℤ) ≅ Hom_{ℤS_{3}}(ℤ, Λ_{sgn}^{∨}),

and a spectral sequence

Ext_{ℤ}^{p}(H_{q}(S_{3}, Λ_{sgn}), ℤ) ⇒ H^{p+q}(S_{3}, Λ_{sgn}^{∨}).  (19)

The input for this spectral sequence is computed in the following lemma.

Lemma 10.4. The coinvariants of Λ and Λ_{sgn} under the S_{3} action are

H_{0}(S_{3}, Λ) = (3, c_{4}, c_{6}) ⊕ ab^{−1}dℤ/3[Δ]
H_{0}(S_{3}, Λ_{sgn}) = d(3, c_{4}, c_{6}) ⊕ ab^{−1}ℤ/3[Δ],

where (3, c_{4}, c_{6}) is the ideal of the ring Λ^{S3} = MF_{∗} of modular forms generated by 3, c_{4} and c_{6}, and d(3, c_{4}, c_{6}) is the corresponding submodule of the free Λ^{S_{3}}- module generated by d.

Proof. We use the exact sequence

0 → Ĥ^{−1}(S_{3}, M) → H_{0}(S_{3}, M) ⟶^{N} H^{0}(S_{3}, M) → Ĥ^{0}(S_{3}, M) → 0. (20)

For M = Λ, this is

0 → ab^{−1}dℤ/3[Δ] → H_{0}(S_{3}, Λ) ⟶^{N} ℤ[c_{4}, c_{6}, Δ]/(∼) ⟶^{π} ℤ/3[Δ] → 0,

where the rightmost map π sends c_{4} and c_{6} to zero, and Δ to Δ. Hence its kernel is the ideal (3, c_{4}, c_{6}), which is a free ℤ-module so we have a splitting as claimed. Similarly, for M = Λ_{sgn}, the exact sequence (20) becomes

0 → ab^{−1}ℤ/3[Δ] → H_{0}(S_{3}, Λ_{sgn}) ⟶^{N} dℤ[c_{4}, c_{6}, Δ]/(∼) ⟶^{dπ} dℤ/3[Δ] → 0.

The kernel of dπ is the ideal d(3, c_{4}, c_{6}), and the result follows.

Corollary 10.5. The S_{3}-invariants of the dual module Λ^{∨} are the module dual to the ideal (3, c_{4}, c_{6}), and the S_{3}-invariants of the dual module Λ_{sgn}^{∨} are the module dual to the ideal d(3, c_{4}, c_{6}).

Proof. In view of the above spectral sequence (19), to compute the invariants it suffices to compute the coinvariants, which we just did in Lemma 10.4, and dualize.

We need one more computational result crucial in the proof of the main Theorem 13.1.

Proposition 10.6. There is an isomorphism of modules over the cohomology ring H^{∗}(S_{3}, π_{∗}T mf(2))

H^{∗}(S_{3}, π_{∗}I_{ℤ}T mf(2)) ≅ H^{∗}(S_{3}, π_{∗}Σ^{21}T mf(2)).

Proof. We need to show that H^{∗}(S_{3}, Σ^{9}Λ_{sgn} ⊕ Λ^{∨}) is a shift by 12 of H^{∗}(S_{3}, Σ^{9}Λ ⊕ Λ_{sgn}^{∨}). First of all, we look at the non-torsion elements. Putting together the results from equation (18) and Corollary 10.5 yields

H^{0}(S_{3}, Σ^{9}Λ_{sgn} ⊕ Λ^{∨}) = dH^{0}(S_{3}, Σ^{9}Λ ⊕ Λ_{sgn}^{∨}),

and indeed we shall find that the shift in higher cohomology also comes from multiplication by the element d (of topological grading 12). Now we look at the higher cohomology groups, computed in Proposition 10.1. Identifying (b^{−1})^{∨} with b, we obtain, in positive cohomological grading

H^{∗}(S_{3}, π_{∗}I_{ℤ}T mf(2)) = Σ^{9}H^{∗}(S_{3}, Λ_{sgn}) ⊕ H^{∗}(Λ^{∨})
= ℤ/3[b^{2}, Δ]⟨Σ^{9}b, Σ^{9}ab, Σ^{9}b^{2}d, Σ^{9}ad⟩
⊕ℤ/3[b^{2}, Δ]/(Δ^{∞})⟨b^{2}Δ, a^{∨}b^{2}Δ, bd, a^{∨}bd⟩,

which we are comparing to

Σ^{21}H^{∗}(S_{3}, π_{∗}T mf(2)) = Σ^{9}dH^{∗}(S_{3}, Λ) ⊕ dH^{∗}(S_{3}, Λ_{sgn}^{∨})
= ℤ/3[b^{2}, Δ]⟨Σ^{9}b^{2}d, Σ^{9}ad, Σ^{9}bΔ, Σ^{9}abΔ⟩
⊕ℤ/3[b^{2}, Δ]/(Δ^{∞})⟨bdΔ, a^{∨}bdΔ, b^{2}Δ, a^{∨}b^{2}Δ⟩.

Everything is straightforwardly identical, except for the match for the generators Σ^{9}b, Σ^{9}ab ∈ Σ^{9}H^{∗}(S_{3}, Λ_{sgn}) which have cohomological gradings 2 and 3, and topological gradings 7 and 10 respectively. On the other side of the equation we have generators a^{∨}bd, bd ∈ H^{∗}(S_{3}, Λ^{∨}) = Σ^{9}H^{∗}(S_{3}, H^{1}(M(2), ω^{∗})), whose cohomological gradings are 2 and 3, and topological 7 and 10 respectively. Identifying these elements gives an isomorphism which is compatible with multiplication by a, b, d.

10.4 Localization

We record the behavior of our group cohomology rings when we invert a modular form; in Section 11.1 we will be inverting c_{4} and Δ.

Proposition 10.7. Let M be one of the modules Λ, Λ_{sgn}; the ring of modular forms MF_{∗} = Λ^{S_{3}} acts on M. Let m ∈ MF_{∗}, and let M[m^{−1}] be the module obtained from M by inverting the action of m. Then

H^{∗}(S_{3}, M[m^{−1}]) ≅ H^{∗}(S_{3}, M)[m^{−1}].

Proof. Since the ring of modular form is S_{3}-invariant, the group S_{3} acts MF_{∗}- linearly on M; in fact, S_{3} acts ℤ[m^{±1}]-linearly on M, where m is our chosen modular form. By Exercise 6.1.2 and Proposition 3.3.10 in [Wei94], it follows that

H^{∗}(S_{3}, M[m^{−1}]) = Ext_{Z[m^{±1}][S_{3}]}^{∗}(ℤ[m^{±1}], M[m^{−1}])
= Ext_{ℤ[m][S_{3}]}^{∗}(ℤ[m], M)[m^{−1}] = H^{∗}(S_{3}, M)[m^{−1}].

Note that if M is one of the dual modules Λ^{∨} or Λ_{sgn}^{∨}, the elements of positive degree (i.e. non-scalar elements) in the ring of modular forms MF_{∗} act on M nilpotently. Therefore M[m^{−1}] = 0 for such an m. Moreover, for degree reasons, c_{4}a = 0 = c_{4}b, and we obtain the following result.

Proposition 10.8. The higher group cohomology of S_{3} with coefficients in π_{∗}T mf(2)[c_{4}^{−1}] vanishes, and

H^{∗}(S_{3}, π_{∗}T mf(2)[c_{4}^{−1}]) = H^{0}(S_{3}, Λ)[c_{4}^{−1}] = MF_{∗}[c_{4}^{−1}].

Inverting Δ has the effect of annihilating the cohomology that comes from the negative homotopy groups of T mf(2); in other words,

H^{∗}(S_{3}, π_{∗}T MF (2)) = H^{∗}(S_{3}, π_{∗}T mf(2)[Δ^{−1}]) = H^{∗}(S_{3}, Λ)[Δ^{−1}].

11 Homotopy Fixed Points

In this section we will use the map q : M(2) → M[1/2] and our knowledge about T mf(2) from the previous sections to compute the homotopy groups of T mf, in a way that displays the self-duality we are looking for. Economizing the notation, we will write M to mean M[1/2] throughout.

11.1 Homotopy Fixed Point Spectral Sequence

We will use Theorem 6.1 to compute the homotopy groups of T mf via the homotopy fixed point spectral sequence

H^{∗}(S_{3}, π_{∗}T mf(2)) ⇒ π_{∗}T mf.  (21)

We will employ two methods of calculating the E_{2}-term of this spectral sequence: the first one is more conducive to computing the differentials, and the second is more conducive to understanding the duality pairing.

Method One

The moduli stack M has an open cover by the substacks M^{0} = M[Δ^{−1}] and M[c_{4}^{−1}], giving the cube of pullbacks

\xymatrix{ & M^{0}[c_{4}^{−1}] \ar[rr] \ar[dd] &  & M^{0} \ar[dd] \\ M(2)^{0}[c_{4}^{−1}] \ar@{-->}[ur] \ar[rr] \ar[dd] &  & M(2)^{0} \ar@{-->}[ur] \ar[dd] \\  & M[c_{4}^{−1}] \ar[rr] &  & M. \\ M(2)[c_{4}^{−1}] \ar@{-->}[ur] \ar[rr] &  & M(2) \ar@{-->}[ur]}

Since Δ, c_{4} are S_{3}-invariant elements of H^{∗}(M(2), ω^{∗}), the maps in this diagram are compatible with the S_{3}-action. Taking global sections of the front square, we obtain a cofiber sequence

T mf(2) → T MF (2) ∨ T mf(2)[c_{4}^{−1}] → T MF (2)[c_{4}^{−1}],

compatible with the S_{3}-action. Consequently, there is a cofiber sequence of the associated homotopy fixed point spectral sequences, converging to the cofiber sequence from the rear pullback square of the above diagram

T mf → T MF ∨ T mf[c_{4}^{−1}] → T MF [c_{4}^{−1}].  (22)

We would like to deduce information about the differentials of the spectral sequence for T mf from the others. According to Proposition 10.8, we know that the spectral sequences for T mf[c_{4}^{−1}] and T MF [c_{4}^{−1}] collapse at their E_{2} pages and that all torsion elements come from T MF. From Proposition 10.1, we know what they are.

As the differentials in the spectral sequence (21) involve the torsion elements in the higher cohomology groups H^{∗}(S_{3}, π_{∗}T mf(2)), they have to come from the spectral sequence for T MF^{5}, where they are (by now) classical. They are determined by the following lemma, which we get from [Bau08] or [Rez01].

Lemma 11.1. The elements α and β in π_{∗}S_{(3)} are mapped to a and bd respectively under the Hurewicz map π_{∗}S_{(3)} → π_{∗}T MF.

Hence, d_{5}(Δ) = αβ^{2}, d_{9}(αΔ) = β^{5}, and the rest of the pattern follows by multiplicativity. The aggregate result is depicted in the chart of Figure 2.

Method Two

On the other hand, we could proceed using our Serre duality for M(2) and the spectral sequence (19). The purpose is to describe the elements below the line t = 0 as elements of H^{∗}(S_{3}, Λ_{sgn}^{∨}). Indeed, according to the Serre duality pairing from Theorem 8.5, we have an isomorphism

H^{s}(S_{3}, H^{1}(M(2), ω^{t−4})) ≅ H^{s}(S_{3}, Λ_{sgn}^{∨}),

and the latter is computed in Corollary 10.5 via the collapsing spectral sequence (19) and Lemma 10.4

Ext_{ℤ}^{p}(H_{q}(S_{3}, Λ_{sgn}), ℤ) ⇒ H^{p+q}(S_{3}, Λ_{sgn}^{∨}).

In Section 10 we computed the input for this spectral sequence. The coinvariants are given as

H_{0}(S_{3}, Λ_{sgn}) = ℤ/3[Δ]ab^{−1} ⊕ d(3, c_{4}, c_{6})

by Lemma 10.4, and the remaining homology groups are computed via the Tate cohomology groups. Namely, for q ≥ 1,

H_{q}(S_{3}, Λ_{sgn}) ≅ Ĥ^{−q−1}(S_{3}, Λ_{sgn})

which, according to Proposition 10.1, equals to the part of cohomological degree (−q − 1) in the Tate cohomology of Λ_{sgn}, which is Rb ⊕ Rd, where R = ℤ/3[a, b^{±2}, Δ]/(a^{2}). Recall, the cohomological grading of a is one, that of b is two, and Δ has cohomological grading zero. In particular, we find that the invariants H^{0}(S_{3}, Λ_{sgn}^{∨}) are the module dual to the ideal d(3, c_{4}, c_{6}). This describes the negatively graded non-torsion elements. For example, the element dual to 3d is in bidegree (t, s) = (−10, 1). We can similarly describe the torsion elements as duals. If X is a torsion abelian group, let X^{∨} denote Ext_{ℤ}^{1}(X, ℤ), and for x ∈ X, let x^{∨} denote the element in X^{∨} corresponding to x under an isomorphism X ≅ X^{∨}. For example, (ab^{−1})^{∨} is in bidegree (−6, 2), and the element corresponding to b^{−2}d lies in bidegree (−10, 5).

5Explicitly, the cofiber sequence (22) gives a commutative square of spectral sequences just as in the proof of Theorem 13.1 below, which allows for comparing differentials.

Documenta Mathematica 17 (2012) 271–311

300 Vesna Stojanoska

Figure 2: Homotopy fixed point spectral sequence (21) for π_{∗}T mf

Documenta Mathematica 17 (2012) 271–311

11.2 Duality Pairing

Consider the non-torsion part of the spectral sequence (21) for π_{∗}T mf. According to Lemma 10.4, on the E_{2} page, it is MF_{∗} ⊕ Σ^{−9}d^{∨}(3, c_{4}, c_{6})^{∨}. Applying the differentials only changes the coefficients of various powers of Δ. Namely, the only differentials supported on the zero-line are d_{5}(Δ^{m}) for non-negative integers m not divisible by 3, which hit a corresponding class of order 3. Therefore, 3^{ϵ}Δ^{m} is a permanent cycle, where ϵ is zero if m is divisible by 3 and one otherwise. In the negatively graded part, only (3Δ^{3k}d)^{∨}, for non-negative k, support a differential d_{9} and hit a class of order 3, thus (3^{ϵ}Δ^{m}d)^{∨} are permanent cycles, for ϵ as above. The pairing at E_{∞} is thus obvious: 3^{ϵ}Δ^{m}c_{4}^{i}c_{6}^{j} and (3^{ϵ}Δ^{m}dc_{4}^{i}c_{6}^{j})^{∨} match up to the generator of ℤ in π_{−21}T mf. The pairing on torsion depends even more on the homotopy theory, and interestingly not only on the differentials, but also on the exotic multiplications by α. The non-negative graded part is ℤ[Δ^{3}] tensored with the pattern in Figure 3, whereas the negative graded part is (ℤ[Δ^{3}]/Δ^{∞})\frac{d^{∨}}{λ_{1}λ_{2}} tensored with the elements depicted in Figure 4.

Figure 3: Torsion in positive degrees

Everything pairs to α^{∨}β^{5}(Δ^{2})^{∨}, which is d_{9}(\frac{(3d)^{∨}}{λ_{1}λ_{2}} ), i.e. the image under d_{9} of 1/3 of the dualizing class. Even though α^{∨}β^{5}(Δ^{2})^{∨} is zero in the homotopy groups of T mf, the corresponding element in the homotopy groups of the K(2)-local sphere is nontrivial [HKM08], thus it makes sense to talk about the pairing.

Figure 4: Torsion in negative degrees 12 The Tate Spectrum

In this section we will relate the duality apparent in the homotopy groups of T mf ≃ T mf(2)^{hS_{3}} to the vanishing of the associated Tate spectrum. The objective is to establish the following:

Theorem 12.1. The norm map T mf(2)_{hS_{3}} → T mf(2)^{hS_{3}} is an equivalence.

A key role is played by the fact that S_{3} has periodic cohomology. Generalized Tate cohomology was first introduced by Adem-Cohen-Dwyer in [ACD89]. However, it was Greenlees and May who generalized and improved the theory, and, more importantly, developed excellent computational tools in [GM95]. In this section, we shall summarize the relevant results from [GM95] and apply them to the problem at hand. Suppose k is a spectrum with an action by a finite group G; in the terminology of equivariant homotopy theory, this is known as a naive G-spectrum. There is a norm map [GM95, 5.3], [LMSM86, II.7.1] from the homotopy orbit spectrum k_{hG} to the homotopy fixed point spectrum k^{hG} whose cofiber we shall call the Tate spectrum associated to the G-spectrum k, and for simplicity denote it by k^{tG}

k_{hG} → k^{hG} → k^{tG}.  (23)

According to [GM95, Proposition 3.5], if k is a ring spectrum, then so are the associated homotopy fixed point and Tate spectra, and the map between them is a ring map. We can compute the homotopy groups of each of the three spectra in (23) using the Atiyah-Hirzebruch-type spectral sequences Ě_{∗}, E_{∗}, Ê_{∗} [GM95, Theorems 10.3, 10.5, 10.6]

Ě_{2}^{p,q} = H_{−p}(G, π_{q}k) ⇒ π_{q−p}k_{hG}
E_{2}^{p,q} = H^{p}(G, π_{q}k) ⇒ π_{q−p}k^{hG}
Ê_{2}^{p,q} = Ĥ^{p}(G, π_{q}k) ⇒ π_{q−p}k^{tG},

which are conditionally convergent. As these spectral sequences can be constructed by filtering EG, the first two are in fact the homotopy fixed point and homotopy orbit spectral sequences. In the case when k = T mf(2), the first two are half-plane spectral sequences, whereas the third one is in fact a full plane spectral sequence. Moreover, the last two are spectral sequences of differential algebras. The norm cofibration sequence (23) relates these three spectral sequences by giving rise to maps between them, which on the E_{2}-terms are precisely the standard long exact sequence in group cohomology:

⋯ → H_{−p}(G, M) → H^{p}(G, M) → Ĥ^{p}(G, M) → H_{−p−1}(G, M) → ⋯ (24)

The map of spectral sequences E_{∗} → Ê_{∗} is compatible with the differential algebra structure, which will be important in our calculations as it will allow us to determine the differentials in the Tate spectral sequence (and then further in the homotopy orbit spectral sequence).

Proof of Theorem 12.1 . We prove that the norm is an equivalence by showing that the associated Tate spectrum is contractible, using the above Tate spectral sequence for the case of k = T mf(2) and G = S_{3}

Ĥ^{p}(S_{3}, π_{2t−q}T mf(2)) = Ĥ^{p}(S_{3}, H^{q}(M(2), ω^{t})) ⇒ π_{2t−p−q}T mf(2)^{tS_{3}}. (25)

The E_{2}-page is the Tate cohomology

Ĥ^{∗}(S_{3}, Λ ⊕ Σ^{−9}Λ_{sgn}^{∨}),

which we computed in Proposition 10.1 to be

R ⊕ Rbd ⊕ \frac{ηd^{∨}}{λ_{1}λ_{2}} R^{∨} ⊕ \frac{ηb^{∨}}{λ_{1}λ_{2}} R^{∨},

where R = ℤ/3[a, b^{±2}, Δ]/(a^{2}), and R^{∨} = Ext_{ℤ}^{1}(R, ℤ). Comparing the two methods for computing the E_{2}-page of the homotopy fixed point spectral sequence (21) identifies \frac{ηd^{∨}}{λ_{1}λ_{2}} with \frac{α}{Δ} and \frac{ηb^{∨}}{λ_{1}λ_{2}} with \frac{α}{β}. Further, we can identify b^{∨} with b^{−1} and similarly for Δ, as it does not change the ring structure, and does not cause ambiguity. We obtain

\frac{ηd^{∨}}{λ_{1}λ_{2}} R^{∨} = \frac{α}{Δ}R^{∨} = \frac{1}{Δ}ℤ/3[a, b^{±2}, Δ^{−1}]/(a^{2})

\frac{ηb^{∨}}{λ_{1}λ_{2}} R^{∨} = \frac{α}{β}R^{∨} = \frac{b}{d}ℤ/3[a, b^{±2}, Δ^{−1}]/(a^{2}) = \frac{β}{Δ}ℤ/3[a, b^{±2}, Δ^{−1}]/(a^{2}).

Summing all up, we get the E_{2}-page of the Tate spectral sequence

Ĥ^{∗}(S_{3}, π_{∗}T mf(2)) = ℤ/3[α, β^{±1}, Δ^{±1}]/(α^{2}),  (26)

depicted in Figure 5. The compatibility with the homotopy fixed point spectral sequence implies that d_{5}(Δ) = αβ^{2} and d_{9}(αΔ^{2}) = β^{5}; by multiplicativity, we obtain a differential pattern as showed below in Figure 5. From the chart we see that the tenth page of the spectral sequence is zero, and, as this was a conditionally convergent spectral sequence, it follows that it is strongly convergent, thus the Tate spectrum T mf(2)^{tS_{3}} is contractible.

12.1 Homotopy Orbits

As a corollary of the vanishing of the Tate spectrum T mf(2)^{tS_{3}}, we fully describe the homotopy orbit spectral sequence

H_{s}(S_{3}, π_{t}T mf(2)) ⇒ π_{t+s}T mf(2)_{hS_{3}} = π_{t+s}T mf.  (27)

Figure 5: Tate spectral sequence (25) for π_{∗}T mf(2)^{tS_{3}}

Duality for Topological Modular Forms 305 Figure 6: Homotopy orbit spectral sequence (27) for π_{∗}T mf

Documenta Mathematica 17 (2012) 271–311

From (26), we obtain the higher homology groups as well as the differentials. If S denotes the ring ℤ/3[β^{−1}, Δ^{±1}],

⨁_{s>0} H_{s}(S_{3}, π_{t}T mf(2)) = Σ^{−1}(β^{−1}S ⊕ αβ^{−2}S).

(The suspension shift is a consequence of the fact that the isomorphism comes from the coboundary map Ĥ^{∗} → H_{−∗−1}.) The coinvariants are computed in Lemma 10.4. The spectral sequence is illustrated in Figure 6, with the the topological grading on the horizontal axis and for consistency, the cohomological on the vertical axis.

13 Duality for T mf

In this section we finally combine the above results to arrive at self-duality for T mf. The major ingredient in the proof is Theorem 12.1, which gives an isomorphism between the values of a right adjoint (homotopy fixed points) and a left adjoint (homotopy orbits). This situation often leads to a Grothendieck-Serre-type duality, which in reality is a statement that a functor (derived global sections) which naturally has a left adjoint (pullback) also has a right adjoint [FHM03]. Consider the following chain of equivalences involving the Anderson dual of T mf

I_{ℤ}T mf = F (T mf, I_{ℤ}) ← F (T mf(2)^{hS_{3}}, I_{ℤ}) → F (T mf(2)_{hS_{3}}, I_{ℤ})
≃ F (T mf(2), I_{ℤ})^{hS_{3}} ≃ (Σ^{9}T mf(2)_{sgn})^{hS_{3}},

which implies a homotopy fixed point spectral sequence converging to the homotopy groups of I_{ℤ}T mf. From our calculations in Section 10 made precise in Proposition 10.6, the E_{2}-term of this spectral sequence is isomorphic to the E_{2}-term for the homotopy fixed point spectral sequence for T mf(2)^{hS_{3}}, shifted by 21 to the right. A shift of 9 comes from the suspension, and an additional shift of 12 comes from the twist by sign (which is realized by multiplication by the element d whose topological degree is 12). It is now plausible that I_{ℤ}T mf might be equivalent to Σ^{21}T mf; it only remains to verify that the differential pattern is as desired. To do this, we use methods similar to the comparison of spectral sequences in [Mil81] and in the algebraic setting, [Del71]: a commutative square of spectral sequences, some of which collapse, allowing for the tracking of differentials.

Theorem 13.1. The Anderson dual of T mf[1/2] is Σ^{21}T mf[1/2].

Remark 13.2. Here again Anderson duals are taken in the category of spectra with 2 inverted, and 2 will implicitly be inverted everywhere in order for the presentation to be more compact. In particular, ℤ will denote ℤ[1/2], and ℚ/ℤ will denote ℚ/ℤ[1/2].

Proof. For brevity, let us introduce the following notation: for R any of ℤ, ℚ, ℚ/ℤ, we let A_{R}^{•} be the cosimplicial spectrum F (ES_{3+} ∧_{S_{3}} T mf(2), I_{R}).

In particular we have that

A_{R}^{h} = F ((S_{3})_{+}^{∧(h+1)} ∧_{S_{3}} T mf(2), I_{R}).

Then the totalization Tot A_{ℤ}^{•} ≃ I_{ℤ}T mf is equivalent to the fiber of the natural map Tot A_{ℚ}^{•} → Tot A_{ℚ/ℤ}^{•}. In other words, we are looking at the diagram

[figure: A_{ℚ}^{0}; A_{ℚ}^{1}; A_{ℚ}^{2} ⋯; A_{ℚ/ℤ}^{0}; A_{ℚ/ℤ}^{1}; A_{ℚ/Z}^{2} ⋯]

and the fact that totalization commutes with taking fibers gives us two ways to compute the homotopy groups of I_{ℤ}T mf. Taking the fibers first gives rise to the homotopy fixed point spectral sequence whose differentials we are to determine: Each vertical diagram gives rise to an Anderson duality spectral sequence (2), which collapses at E_{2}, as the homotopy groups of each ((S_{3})_{+}^{∧(h+1)} ∧_{S_{3}} T mf(2))

are free over ℤ. On the other hand, assembling the horizontal direction first gives a map of the ℚ and ℚ/ℤ-duals of the homotopy fixed point spectral sequence for the S_{3}-action on T mf(2); this is because ℚ and ℚ/ℤ are injective ℤ-modules, thus dualizing is an exact functor. Let R^{•} denote the standard injective resolution of ℤ, namely R^{0} = ℚ and R^{1} = ℚ/ℤ related by the obvious quotient map. Then, schematically, we have a diagram of E_{1}-pages

[figure: Hom_{ℤ} (ℤ[S_{3}]^{⊗(h+1)} ⊗_{ℤ[S_{3}]} π_{t}T mf(2), R^{v}); A; B; Hom_{ℤ} (π_{t+h}T mf(2)_{hS_{3}}, R^{v}); D; Hom_{ℤ} (ℤ[S_{3}]^{⊗(h+1)} ⊗_{ℤ[S_{3}]} π_{t}T mf(2), ℤ); C; π_{−t−h−v}I_{ℤ}T mf(2)_{hS_{3}}.]

The spectral sequence A collapses at E_{2}, and C is the homotopy fixed point spectral sequence that we are interested in: its E_{2} page is the S_{3}-cohomology of the ℤ-duals of π_{∗}T mf(2), which are the homotopy groups of the Anderson dual of T mf(2)

H^{∗} Hom_{ℤ} (ℤ[S_{3}]^{⊗(h+1)} ⊗_{ℤ[S_{3}]} π_{t}T mf(2), ℤ)

≅ H^{∗} Hom_{ℤ[S_{3}]} (ℤ[S_{3}]^{⊗(h+1)}, Hom_{ℤ}(π_{t}T mf(2), ℤ))

≅ H^{h}(S_{3}, π_{−t}I_{ℤ}T mf(2)).

Indeed, the E_{2}-pages assemble in the following diagram

[figure: Ext_{ℤ}^{v} (H_{h}(S_{3}, π_{t}T mf(2)), ℤ); A; B; Ext_{ℤ}^{v} (π_{t+h}T mf(2)_{hS_{3}}, ℤ); D; H^{h+v}(S_{3}, π_{−t}I_{ℤ}T mf(2)); C; π_{−t−h−v}I_{ℤ}T mf(2)_{hS_{3}}.]

The spectral sequence A is the dual module group cohomology spectral sequence (19), and it collapses, whereas D is the Anderson duality spectral sequence (2) which likewise collapses at E_{2}. The spectral sequence B is dual to the homotopy orbit spectral sequence Ě_{∗} (27), which we have completely described in Figure 6. Now [Del71, Proposition 1.3.2] tells us that the differentials in B are compatible with the filtration giving C if and only if A collapses, which holds in our case. Consequently, B and C are isomorphic. In conclusion, we read off the differentials from the homotopy orbit spectral sequence (Figure 6). There are only non-zero d_{5} and d_{9}. For example, the generators in degrees (6, 3) and (2, 7) support a d_{5} as the corresponding elements in (27) are hit by a differential d_{5}. Similarly, the generator in (1, 72) supports a d_{9}, as it corresponds to an element hit by a d_{9}. There is no possibility for any other differentials, and the chart is isomorphic to a shift by 21 of the one in Figure 2. By now we have an abstract isomorphism of the homotopy groups of I_{ℤ}T mf and Σ^{21}T mf, as π_{∗}T mf-modules. As in Theorem 9.1, we build a map realizing this isomorphism by specifying the dualizing class and then extending using the T mf-module structure on I_{ℤ}T mf.

As a corollary, we recover [Beh06, Proposition 2.4.1].

Corollary 13.3. At odd primes, the Gross-Hopkins dual of L_{K(2)}T mf is Σ^{22}L_{K(2)}T mf.

Proof. The spectrum T mf is E(2)-local, hence we can compute the Gross-Hopkins dual I_{2}T mf as ΣL_{K(2)}I_{ℤ}T mf by (4).

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Vesna Stojanoska

Massachusetts Institute of Technology Cambridge MA 02139 USA 312

Documenta Mathematica 17 (2012)