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.