Inventiones mathematicae

, Volume 203, Issue 2, pp 359–416 | Cite as

Topological modular forms with level structure

  • Michael Hill
  • Tyler LawsonEmail author


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.



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.


  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)CrossRefMathSciNetzbMATHGoogle 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)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Ando, M.: Power operations in elliptic cohomology and representations of loop groups. Trans. Am. Math. Soc. 352(12), 5619–5666 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Baker, A.: Hecke operators as operations in elliptic cohomology. J. Pure Appl. Algebra 63(1), 1–11 (1990)CrossRefMathSciNetzbMATHGoogle 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)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Behrens, M., Lawson, T.: Topological automorphic forms. Mem. Am. Math. Soc. 204(958), xxiv+141 (2010)MathSciNetzbMATHGoogle 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)Google Scholar
  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)CrossRefMathSciNetzbMATHGoogle 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)CrossRefMathSciNetzbMATHGoogle 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. 143–316Google Scholar
  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)Google Scholar
  15. 15.
    Ganter, N.: Power operations in orbifold Tate \(K\)-theory. Homol. Homotopy Appl. 15(1), 313–342 (2013)CrossRefMathSciNetzbMATHGoogle 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)Google Scholar
  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)Google Scholar
  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)Google Scholar
  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)CrossRefMathSciNetzbMATHGoogle 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)Google Scholar
  23. 23.
    Jardine, J.F.: Presheaves of symmetric spectra. J. Pure Appl. Algebra 150(2), 137–154 (2000)CrossRefMathSciNetzbMATHGoogle 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)CrossRefMathSciNetzbMATHGoogle 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)Google Scholar
  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)Google Scholar
  29. 29.
    Laures, G.: \(K(1)\)-local topological modular forms. Invent. Math. 157(2), 371–403 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Lawson, T., Naumann, N.: Commutativity conditions for truncated Brown-Peterson spectra of height 2. J. Topol. 5(1), 137–168 (2012)CrossRefMathSciNetzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)Google Scholar
  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)Google Scholar
  35. 35.
    Lurie, J.: Higher Topos Theory, Annals of Mathematics Studies, vol. 170. Princeton University Press, Princeton (2009)zbMATHGoogle 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)Google Scholar
  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)CrossRefMathSciNetzbMATHGoogle 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)Google Scholar
  41. 41.
    Nizioł, W.: \(K\)-theory of log-schemes. I. Doc. Math. 13, 505–551 (2008)MathSciNetzbMATHGoogle 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)CrossRefMathSciNetzbMATHGoogle 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)CrossRefMathSciNetzbMATHGoogle 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)Google Scholar
  46. 46.
    Robinson, A., Whitehouse, S.: Operads and \(\Gamma \)-homology of commutative rings. Math. Proc. Camb. Philos. Soc. 132(2), 197–234 (2002)CrossRefMathSciNetzbMATHGoogle 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)Google Scholar
  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)Google Scholar
  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)Google Scholar
  50. 50.
    Silverman, J.H.: The arithmetic of elliptic curves, second ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht (2009)Google Scholar
  51. 51.
    Stojanoska, V.: Duality for topological modular forms. Doc. Math. 17, 271–311 (2012)MathSciNetzbMATHGoogle Scholar
  52. 52.
    White, D.: Model structures on commutative monoids in general model categories, arXiv:1403.6759

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematics (617) 271-7062University of VirginiaCharlottesvilleUSA
  2. 2.Department of MathematicsUniversity of MinnesotaMinneapolisUSA

Personalised recommendations