Inventiones mathematicae

, Volume 203, Issue 2, pp 359–416

Topological modular forms with level structure


DOI: 10.1007/s00222-015-0589-5

Cite this article as:
Hill, M. & Lawson, T. Invent. math. (2016) 203: 359. doi:10.1007/s00222-015-0589-5


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.

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