Topological modular forms with level structure


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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  1. 1.

    Ando, M., Hopkins, M.J., Rezk, C.: Multiplicative orientations of KO-theory and of the spectrum of topological modular forms. Preprint, available at:

  2. 2.

    Ando, M., Hopkins, M.J., Strickland, N.P.: Elliptic spectra, the Witten genus and the theorem of the cube. Invent. Math. 146(3), 595–687 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. 3.

    Ando, M., Hopkins, M.J., Strickland, N.P.: The sigma orientation is an \(H_\infty \) map. Am. J. Math. 126(2), 247–334 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. 4.

    Ando, M.: Power operations in elliptic cohomology and representations of loop groups. Trans. Am. Math. Soc. 352(12), 5619–5666 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. 5.

    Baker, A.: Hecke operators as operations in elliptic cohomology. J. Pure Appl. Algebra 63(1), 1–11 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. 6.

    Behrens, M.: Notes on the construction of \(tmf\). available at:

  7. 7.

    Behrens, M.: Buildings, elliptic curves, and the \(K(2)\)-local sphere. Am. J. Math. 129(6), 1513–1563 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. 8.

    Behrens, M., Lawson, T.: Topological automorphic forms. Mem. Am. Math. Soc. 204(958), xxiv+141 (2010)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Basterra, M., Richter, B.: (Co-)homology theories for commutative (\(S\)-)algebras. In: Structured ring spectra, London Math. Soc. Lecture Note Ser., vol. 315, Cambridge Univ. Press, Cambridge, pp. 115–131 (2004)

  10. 10.

    Baker, A., Richter, B.: Realizability of algebraic Galois extensions by strictly commutative ring spectra. Trans. Am. Math. Soc. 359(2), 827–857 (2007). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  11. 11.

    Buzzard, K.: Computing weight one modular forms over \(\mathbb{C}\) and \(\overline{\mathbb{F}}_p\), arXiv:1205.5077

  12. 12.

    Dugger, D., Hollander, S., Isaksen, D.C.: Hypercovers and simplicial presheaves. Math. Proc. Camb. Philos. Soc. 136(1), 9–51 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. 13.

    Deligne, P., Rapoport, M.: Les schémas de modules de courbes elliptiques. In: Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math., vol. 349, Springer, Berlin, 1973, pp. 143–316

  14. 14.

    Faltings, G., Chai, C.-L.: Degeneration of abelian varieties. In: Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 22, Springer-Verlag, Berlin (1990) (With an appendix by David Mumford)

  15. 15.

    Ganter, N.: Power operations in orbifold Tate \(K\)-theory. Homol. Homotopy Appl. 15(1), 313–342 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. 16.

    Goerss, P.G., Hopkins, M.J.: Moduli spaces of commutative ring spectra. Structured ring spectra, London Math. Soc. Lecture Note Ser., vol. 315, Cambridge Univ. Press, Cambridge, pp. 151–200 (2004)

  17. 17.

    Goerss, P.G.: Realizing families of Landweber exact homology theories. New topological contexts for Galois theory and algebraic geometry (BIRS 2008), Geom. Topol. Monogr., vol. 16, Geom. Topol. Publ. Coventry, 49–78 (2009)

  18. 18.

    Hartshorne, R.: Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin (1966)

  19. 19.

    Hopkins, M.J., Mahowald, M.: From elliptic curves to homotopy theory. Preprint,

  20. 20.

    Hopkins, M.J.: \(K(1)\)-local \(E_\infty \)-ring spectra. Preprint, available at:

  21. 21.

    Hovey, M., Shipley, B., Smith, J.: Symmetric spectra. J. Am. Math. Soc. 13(1), 149–208 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. 22.

    Illusie, L.: An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology. Astérisque, Cohomologies \(p\)-adiques et applications arithmétiques, II, no. 279, pp. 271–322 (2002)

  23. 23.

    Jardine, J.F.: Presheaves of symmetric spectra. J. Pure Appl. Algebra 150(2), 137–154 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. 24.

    Johnstone, P.T.: Sketches of an Elephant: a Topos Theory Compendium. Vol. 2, Oxford Logic Guides, vol. 44. The Clarendon Press, Oxford University Press, Oxford (2002)

    Google Scholar 

  25. 25.

    Katz, N.M.: Higher congruences between modular forms. Ann. of Math. (2) 101, 332–367 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  26. 26.

    Kato, K.: Logarithmic structures of Fontaine-Illusie. Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD pp. 191–224 (1989)

  27. 27.

    Katz, N.M., Mazur, B.: Arithmetic Moduli of Elliptic Curves, Annals of Mathematics Studies, vol. 108. Princeton University Press, Princeton (1985)

    Google Scholar 

  28. 28.

    Landweber, P.S. (ed.): Elliptic curves and modular forms in algebraic topology. Lecture Notes in Mathematics, vol. 1326. Springer-Verlag, Berlin (1988)

  29. 29.

    Laures, G.: \(K(1)\)-local topological modular forms. Invent. Math. 157(2), 371–403 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. 30.

    Lawson, T., Naumann, N.: Commutativity conditions for truncated Brown-Peterson spectra of height 2. J. Topol. 5(1), 137–168 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. 31.

    Lawson, T., Naumann, N.: Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2. Int. Math. Res. Not. 2014(10), 2773–2813 (2014)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Landweber, P.S., Ravenel, D.C., Stong, R.E.: Periodic cohomology theories defined by elliptic curves. The Čech centennial (Boston, MA, 1993), Contemp. Math., vol. 181, Am. Math. Soc., Providence, RI, pp. 317–337 (1995)

  33. 33.

    Lurie, J.: Higher algebra, Draft version available at:

  34. 34.

    Lurie, J.: A survey of elliptic cohomology. Algebraic Topology, Abel Symp., vol. 4, Springer, Berlin, pp. 219–277 (2009)

  35. 35.

    Lurie, J.: Higher Topos Theory, Annals of Mathematics Studies, vol. 170. Princeton University Press, Princeton (2009)

    MATH  Google Scholar 

  36. 36.

    Mathew, A.: The homology of tmf, arXiv:1305.6100

  37. 37.

    Meier, L.: United elliptic homology, Ph.D. thesis, Universität Bonn (2012)

  38. 38.

    Mathew, A., Meier, L.: Affineness and chromatic homotopy theory, arXiv:1311.0514

  39. 39.

    Morava, J.: Forms of \(K\)-theory. Math. Z. 201(3), 401–428 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  40. 40.

    Mahowald, M., Rezk, C.:Topological modular forms of level 3, Pure Appl. Math. Q. 5, no. 2, Special Issue: In honor of Friedrich Hirzebruch. Part 1, 853–872 (2009)

  41. 41.

    Nizioł, W.: \(K\)-theory of log-schemes. I. Doc. Math. 13, 505–551 (2008)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Ochanine, S.: Elliptic genera, modular forms over \(K{\rm O}_*\) and the Brown–Kervaire invariant. Math. Z. 206(2), 277–291 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  43. 43.

    Ogus, A.: Lectures on logarithmic geometry, Draft version available at:

  44. 44.

    Quillen, D.: On the formal group laws of unoriented and complex cobordism theory. Bull. Am. Math. Soc. 75, 1293–1298 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  45. 45.

    Rognes, J.: Topological logarithmic structures, New topological contexts for Galois theory and algebraic geometry (BIRS 2008), Geom. Topol. Monogr., vol. 16, Geom. Topol. Publ., Coventry, pp. 401–544 (2009)

  46. 46.

    Robinson, A., Whitehouse, S.: Operads and \(\Gamma \)-homology of commutative rings. Math. Proc. Camb. Philos. Soc. 132(2), 197–234 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  47. 47.

    Schaeffer, G.J.: The Hecke Stability Method and Ethereal Forms. ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)-University of California, Berkeley (2012)

  48. 48.

    Grothendieck, A.: Cohomologie \(l\)-adique et fonctions \(L\). Lecture Notes in Mathematics, Séminaire de Géometrie Algébrique du Bois-Marie 1965–1966 (SGA 5), Edité par Luc Illusie, vol. 589, Springer-Verlag, Berlin, (1977)

  49. 49.

    Shipley, B.: A convenient model category for commutative ring spectra. Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic \(K\)-theory, Contemp. Math., vol. 346, Am. Math. Soc., Providence, RI, pp. 473–483 (2004)

  50. 50.

    Silverman, J.H.: The arithmetic of elliptic curves, second ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht (2009)

  51. 51.

    Stojanoska, V.: Duality for topological modular forms. Doc. Math. 17, 271–311 (2012)

    MathSciNet  MATH  Google Scholar 

  52. 52.

    White, D.: Model structures on commutative monoids in general model categories, arXiv:1403.6759

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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.

Additional information

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

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

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.


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.


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.


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.


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.


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).

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