Topological modular forms with level structure

Abstract

The cohomology theory known as \(\mathrm{Tmf}\), for “topological modular forms,” is a universal object mapping out to elliptic cohomology theories, and its coefficient ring is closely connected to the classical ring of modular forms. We extend this to a functorial family of objects corresponding to elliptic curves with level structure and modular forms on them. Along the way, we produce a natural way to restrict to the cusps, providing multiplicative maps from \(\mathrm{Tmf}\) with level structure to forms of \(K\)-theory. In particular, this allows us to construct a connective spectrum \(\mathrm{tmf}_0(3)\) consistent with properties suggested by Mahowald and Rezk. This is accomplished using the machinery of logarithmic structures. We construct a presheaf of locally even-periodic elliptic cohomology theories, equipped with highly structured multiplication, on the log-étale site of the moduli of elliptic curves. Evaluating this presheaf on modular curves produces \(\mathrm{Tmf}\) with level structure.

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Acknowledgments

The authors would like to thank Matthew Ando, Mark Behrens, Andrew Blumberg, Scott Carnahan, Jordan Ellenberg, Paul Goerss, Mike Hopkins, Nitu Kitchloo, Michael Mandell, Akhil Mathew, Lennart Meier, Niko Naumann, William Messing, Arthur Ogus, Kyle Ormsby, Charles Rezk, Andrew Salch, George Schaeffer, and Vesna Stojanoska for discussions related to this paper. The anonymous referee of [31] also asked a critical question about compatibility with \(\mathbb {Z}/2\)-actions, motivating our proof that evaluation at the cusp is possible. The ideas in this paper would not have existed without the Loen conference “\(p\)-Adic Geometry and Homotopy Theory” introducing us to logarithmic structures in 2009; the authors would like to thank the participants there, as well as Clark Barwick and John Rognes for organizing it. This paper is written in dedication to Mark Mahowald.

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Correspondence to Tyler Lawson.

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M. Hill was partially supported by NSF DMS–0906285, DARPA FA9550–07–1–0555, and the Sloan foundation. Tyler Lawson was partially supported by NSF DMS–1206008 and the Sloan foundation.

Appendix: The Witten genus

Appendix: The Witten genus

The goal of this section is to construct a map of commutative ring spectra

$$\begin{aligned} \mathrm{tmf}\rightarrow \textit{KO}[\![{q}]\!] \end{aligned}$$

which, on homotopy groups, factors the Witten genus \(MSpin_* \rightarrow \mathbb {Z}[\![{q}]\!]\). Here the power series notation \(\textit{KO}[\![{q}]\!]\) is shorthand for the homotopy limit of the monoid algebras

$$\begin{aligned} \mathop {\mathrm{holim}}_r \textit{KO} \mathop {\wedge }_{} \{1, q, \ldots , q^{r-1}\}_+ \end{aligned}$$

where \(q^{r}\) is identified with the basepoint (as in Sect. 5.1). The main result (Theorem 6.12) is well-known and featured prominently in earlier, unpublished, constructions of \(\mathrm{tmf}\), but to the knowledge of the authors it does not appear in the literature. The relation of the Tate curve to power operations has been extensively explored, especially in this context by Baker [5], Ando [4], Ando–Hopkins–Strickland [2], and Ganter [15].

Definition 6.5

For a chosen prime \(p\), the \(p\)-adic \(K\)-theory of a spectrum \(X\) is

$$\begin{aligned} K^\vee _*(X) = \pi _* L_{K(1)}(K \wedge X) = \pi _* \mathop {\mathrm{holim}}_k (K/p^k \wedge X). \end{aligned}$$

In particular, the coefficient ring \(K^\vee _*\) is the graded ring \(\mathbb {Z}_p[\beta ^{\pm 1}]\).

Here \(K/p^k\) is the mapping cone of the multiplication-by-\(p^k\) endomorphism of \(K\), having a long exact sequence

$$\begin{aligned} \cdots \rightarrow K^\vee _*(X) \mathop {\rightarrow }^{p^k} K^\vee _*(X) \rightarrow \pi _* (K/p^k \mathop {\wedge }_{} X) \rightarrow \cdots \end{aligned}$$
(6.1)

which is natural in \(X\).

Remark 6.6

As \(K\)-modules and \(\textit{KO}\)-modules are automatically \(E(1)\)-local, \(K(1)\)-localizations and \(p\)-completions are equivalent on them.

We recall the following result, which was classically used as a definition of \(K(1)\)-local \(\mathrm{tmf}\) at the prime \(2\).

Proposition 6.7

([20, 29]) At \(p=2\), there are homotopy pushout diagrams

in the category of \(K(1)\)-local commutative ring spectra. Here \(\zeta \) is a topological generator of \(\pi _{-1} L_{K(1)}\mathbb {S} \cong \mathbb {Z}_2\); \(f\) is an element in \(\pi _0 T_\zeta \); and \(h(x)\) is a \(2\)-adically convergent power series such that for any \(K(1)\)-local elliptic commutative ring spectrum \(E\), any map of commutative ring spectra \(T_\zeta \rightarrow E\) automatically sends \(\theta (f)\) and \(h(f)\) to the same element.

We first need to identify the \(p\)-adic \(K\)-theory of \(\textit{KO}[\![{q}]\!]\).

Proposition 6.8

For any prime \(p\), the \(p\)-adic \(K\)-theory of \(\textit{KO}[\![{q}]\!]\) is the ring

$$\begin{aligned} K^\vee _* (\textit{KO}[\![{q}]\!]) \cong \mathrm{Map}_c(\mathbb {Z}_p^\times , K^\vee _*[\![{q}]\!])^{\{\pm 1\}}. \end{aligned}$$

Here the group \(\{\pm 1\}\subset \mathbb {Z}_p^\times \) acts by conjugation on the group of continuous homomorphisms, and the ring \(K^\vee _*[\![{q}]\!]\) is given the \(p\)-adic topology.

This is the universal \(p\)-complete \(\mathbb {Z}[\![{q}]\!]\)-algebra with an isomorphism class of pairs of an invariant \(1\)-form on the Tate curve \(T\) and an identification \(\widehat{\mathbb {G}}_m \mathop {\mathop {\rightarrow }}\limits ^{\sim } \widehat{T}\) between the formal multiplicative group and the formal group of the Tate curve. The map \(V \rightarrow K^\vee _0(\textit{KO}[\![{q}]\!])\) determined by this is a map of \(\psi \text{- }\theta \)-algebras.

Proof

We recall from [20] or [1, 9.2] that the map of \(\psi \text{- }\theta \)-algebras

$$\begin{aligned} K^\vee _* \textit{KO} \rightarrow K^\vee _* K \end{aligned}$$

is the inclusion

$$\begin{aligned} \mathrm{Map}_c(\mathbb {Z}_p^\times , K^\vee _*)^{\{\pm 1\}} \hookrightarrow \mathrm{Map}_c(\mathbb {Z}_p^\times , K^\vee _*) \end{aligned}$$

of sets of continuous maps. The long exact sequence (6.1) gives an identification

$$\begin{aligned} \pi _* (K/p^k \wedge \textit{KO})&\cong \mathrm{Map}_c(\mathbb {Z}_p^\times , (K_*)/p^k)^{\{\pm 1\}}\\&= \mathop {\mathrm{colim}}_{m} \mathrm{Map}((\mathbb {Z}/p^m)^\times , (K_*)/p^k)^{\{\pm 1\}}. \end{aligned}$$

The graded \(\pi _* \textit{KO}\)-module \(\pi _* (\textit{KO}[\![{q}]\!]) \cong \pi _* \textit{KO} \otimes _{\mathbb {Z}} \mathbb {Z}[\![{q}]\!]\) is flat, and so the isomorphism

$$\begin{aligned} K/p^k \wedge \textit{KO}[\![{q}]\!] \cong (K/p^k \wedge \textit{KO}) \mathop {\wedge }_{\textit{KO}} \textit{KO}[\![{q}]\!] \end{aligned}$$

can be re-expressed as an isomorphism

$$\begin{aligned} \pi _*(K/p^k \wedge \textit{KO}[\![{q}]\!]) \cong \mathrm{Map}_c(\mathbb {Z}_p^\times , K_*[\![{q}]\!]/p^k)^{\{\pm 1\}} \end{aligned}$$

by the Künneth formula. Taking limits gives the desired formula for the \(p\)-adic \(K\)-theory.

The action of the group \(\mathbb {Z}_p^\times \) on \(\mathrm{Map}_c(\mathbb {Z}_p^\times , K^\vee _*)^{\{\pm 1\}}\), by premultiplication on the source, is compatible with this isomorphism, and therefore determines the action of \(\mathbb {Z}_p^\times \) on \(K^\vee _*(\textit{KO}[\![{q}]\!])\): it is coinduced from the action of the subgroup \(\{\pm 1\}\) on \(K^\vee _*[\![{q}]\!]\). The ring \(\mathrm{Map}_c(\mathbb {Z}_p^\times , K^\vee _*[\![{q}]\!])\) is the universal ring classifying isomorphisms \(\widehat{\mathbb {G}}_m \rightarrow \widehat{T}\) together with a choice of invariant \(1\)-form, as any such isomorphism differs from the canonical one by a locally constant function to \(\mathbb {Z}_p^\times \). As a \(\{\pm 1\}\)-equivariant algebra over the ring of invariants \(K^\vee _* \textit{KO}[\![{q}]\!]\) it is isomorphic to \(K^\vee _* \textit{KO}[\![{q}]\!] \times K^\vee _* \textit{KO}[\![{q}]\!]\), and so the ring of invariants classifies the quotient by \(Aut(T) \cong \{\pm 1\}\). There is a map \(V \rightarrow K^\vee _0 \textit{KO}[\![{q}]\!]\) determined by the universal property of \(V\).

The element \(q \in K^\vee _0 \textit{KO}[\![{q}]\!]\), since it lifts to an element in \(\pi _0 \textit{KO}[\![{q}]\!]\), is acted on trivially by \(\mathbb {Z}_p^\times \). Moreover, the extended power operation \(\psi ^p\) lifts to the corresponding power operation on the discrete monoid \(\mathbb {N}\), which sends \(q\) to \(q^p\). The resulting map on \(\mathbb {Z}[\![{q}]\!]\) classifies the quotient of the Tate curve by the canonical subgroup \(\mu _p\) of its formal group (Sect. 3.4), and thus the map \(V \rightarrow K^\vee _0(\textit{KO}[\![{q}]\!])\) preserves the operation \(\psi ^p\) (and hence \(\theta \)). \(\square \)

Proposition 6.9

At the prime \(2\), there exists a map of \(K(1)\)-local commutative ring spectra

$$\begin{aligned} L_{K(1)} \mathrm{tmf}\rightarrow L_{K(1)} \textit{KO}[\![{q}]\!] \end{aligned}$$

which, on \(2\)-adic \(K\)-homology, induces the map

$$\begin{aligned} V \rightarrow K^\vee _0(\textit{KO}[\![{q}]\!]) \end{aligned}$$

from Proposition 6.8.

Proof

We will use the description of \(K(1)\)-local \(\mathrm{tmf}\) from Proposition 6.7 to construct this map.

As \(\textit{KO}[\![{q}]\!]^\wedge _2\) is the \(K(1)\)-localization of \(\textit{KO}[\![{q}]\!]\) and has trivial \(\pi _{-1}\), we have a map of commutative ring spectra \(T_\zeta \rightarrow \textit{KO}[\![{q}]\!]^\wedge _2\). The composite map \(T_\zeta \rightarrow K[\![{q}]\!]^\wedge _2\) detects the effect on \(\pi _0\), and is a map to an elliptic cohomology theory, where the latter carries the Tate curve over the power series ring \(\mathbb {Z}[\![{q}]\!]^\wedge _2\). Therefore, the element \(\theta (f) - h(f)\) automatically maps to zero, and we obtain an extension \(L_{K(1)}\mathrm{tmf}\rightarrow L_{K(1)}\textit{KO}[\![{q}]\!]\). \(\square \)

Proposition 6.10

At any odd prime \(p\), there exists a map of \(K(1)\)-local commutative ring spectra

$$\begin{aligned} L_{K(1)} \mathrm{tmf}\rightarrow L_{K(1)} \textit{KO}[\![{q}]\!] \end{aligned}$$

which, on \(p\)-adic \(K\)-theory, induces the map

$$\begin{aligned} V \rightarrow K^\vee _0(\textit{KO}[\![{q}]\!]) \end{aligned}$$

from Proposition 6.8.

Proof

As \(\textit{KO}[\![{q}]\!]\) is the homotopy fixed-point spectrum of the action of \(\{\pm 1\}\) on \(K[\![{q}]\!]\), we have an equivalence

$$\begin{aligned}&\mathrm{Map}_{comm}(L_{K(1)} \mathrm{tmf}, L_{K(1)} \textit{KO}[\![{q}]\!])\\&\quad \simeq \mathrm{Map}_{comm}(L_{K(1)} \mathrm{tmf}, L_{K(1)} K[\![{q}]\!])^{h\{\pm 1\}}. \end{aligned}$$

The Goerss–Hopkins obstruction theory computing this space of maps of \(K(1)\)-local commutative ring spectra produces obstructions in André–Quillen cohomology groups. There is a fringed spectral sequence with \(E_2\)-term given by

$$\begin{aligned} E_2^{s,t} = {\left\{ \begin{array}{ll} \mathrm{Hom}_{\psi \text{- }\theta \text{- }alg/K^\vee _*} (K^\vee _* \mathrm{tmf}, K^\vee _* K[\![{q}]\!])&{}\quad \text {if }(s,t)=(0,0),\\ H^s_{\psi \text{- }\theta \text{- }alg/K^\vee _*} (K^\vee _* \mathrm{tmf}, \Omega ^t K^\vee _* K[\![{q}]\!])&{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

This spectral sequence converges to \(\pi _{t-s} \mathrm{Map}_{comm}(L_{K(1)} \mathrm{tmf}, L_{K(1)} K[\![{q}]\!])\). By [6, 7.5], the fact that \(V\) is formally smooth over \(\mathbb {Z}_p\) implies that the obstruction groups

$$\begin{aligned} H^s_{\psi \text{- }\theta \text{- }alg/K^\vee _*} (K^\vee _* \mathrm{tmf}, \Omega ^t K^\vee _* K[\![{q}]\!]) \end{aligned}$$

are trivial for \(s > 1\) or \(t = 0\).

In particular, the homotopy groups \(\pi _{t} \mathrm{Map}_{comm}(L_{K(1)} \mathrm{tmf}, L_{K(1)} K[\![{q}]\!])\) are \(p\)-adically complete abelian groups for any choice of basepoint, and so the homotopy fixed-point spectral sequence for the action of the group \(\{\pm 1\}\) degenerates. We find that the set of path components is

$$\begin{aligned} \pi _0 \mathrm{Map}_{comm}(L_{K(1)} \mathrm{tmf}, L_{K(1)} K[\![{q}]\!]) \cong \mathrm{Hom}_{\psi \text{- }\theta \text{- }alg/K^\vee _*}(V, K_*[\![{q}]\!])^{\{\pm 1\}}, \end{aligned}$$

and so the map of Proposition 6.8 has a lift which is unique up to homotopy.

\(\square \)

Proposition 6.11

There exists a map of rational commutative ring spectra

$$\begin{aligned} \mathrm{tmf}_{\mathbb {Q}} \rightarrow (\textit{KO}[\![{q}]\!])_{\mathbb {Q}} \end{aligned}$$

which, on homotopy groups, is given by a map

$$\begin{aligned} \mathbb {Q}[c_4, c_6] \rightarrow \mathbb {Q} \otimes \mathbb {Z}[\![{q}]\!][\beta ^{\pm 2}] \end{aligned}$$

sending \(c_4\) and \(c_6\) to their \(q\)-expansions. The two maps

$$\begin{aligned} \mathrm{tmf}\rightarrow \left( \prod _p \textit{KO}[\![{q}]\!]^\wedge _p\right) _{\mathbb {Q}}, \end{aligned}$$

induced by this map and the maps constructed in Propositions 6.9 and 6.10, are homotopic as maps of commutative ring spectra.

Proof

The elements \(c_4\) and \(c_6\) can be realized as maps \(S^8 \rightarrow \mathrm{tmf}_{\mathbb {Q}}\) and \(S^{12} \rightarrow \mathrm{tmf}_{\mathbb {Q}}\) respectively. The induced map of commutative ring spectra \(\mathbb {P}_{\mathbb {Q}}(S^8 \vee S^{12}) \rightarrow \mathrm{tmf}_{\mathbb {Q}}\) is a weak equivalence, and so homotopy classes of commutative ring spectrum maps \(\mathrm{tmf}_{\mathbb {Q}} \rightarrow (\textit{KO}[\![{q}]\!])_{\mathbb {Q}}\) are defined uniquely, up to homotopy, by specifying the images of \(c_4\) and \(c_6\).

Homotopy classes of maps of commutative ring spectra \(\mathrm{tmf}\rightarrow (\prod _p \textit{KO}[\![{q}]\!]^\wedge _p)_{\mathbb {Q}}\) are the same as maps \(\mathrm{tmf}_{\mathbb {Q}} \rightarrow (\prod _p \textit{KO}[\![{q}]\!]^\wedge _p)_{\mathbb {Q}}\), and are similarly determined by the images of \(c_4\) and \(c_6\). Therefore, as the \(K(1)\)-local and rational constructions are both obtained by \(q\)-expansion in a neighborhood of the Tate curve, the resulting pair of maps are homotopic as maps of commutative ring spectra. \(\square \)

Theorem 6.12

There exists a map of commutative ring spectra

$$\begin{aligned} \mathrm{tmf}\rightarrow \textit{KO}[\![{q}]\!] \end{aligned}$$

compatible with the \(K(1)\)-local and rational maps constructed in Propositions 6.9, 6.10, and 6.11.

Proof

We can express the spectrum \(\textit{KO}[\![{q}]\!]\) as a homotopy pullback in the following arithmetic square of commutative ring spectra:

However, from Propositions 6.9, 6.10, and 6.11 we obtain maps from \(\mathrm{tmf}\) to the rational and \(p\)-completed entries which are homotopic, and therefore a map from \(\mathrm{tmf}\) to the homotopy pullback. \(\square \)

Remark 6.13

As the spectrum \((\prod _p \textit{KO}[\![{q}]\!]^\wedge _p)_{\mathbb {Q}}\) has trivial homotopy groups in degrees \(9\) and \(13\), the path components of the mapping space

$$\begin{aligned} \mathrm{Map}_{}\left( \mathrm{tmf},\left( \prod _p \textit{KO}[\![{q}]\!]^\wedge _p\right) _{\mathbb {Q}}\right) \end{aligned}$$

are all simply connected. The Mayer–Vietoris square of mapping spaces shows that there is a unique homotopy class of map of commutative ring spectra from \(\mathrm{tmf}\) to the pullback.

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Hill, M., Lawson, T. Topological modular forms with level structure. Invent. math. 203, 359–416 (2016). https://doi.org/10.1007/s00222-015-0589-5

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