Topological modular forms with level structure

TL;DR

This paper extends the topological modular forms (Tmf) cohomology theory to include level structures on elliptic curves, introduces a sheaf of elliptic cohomology theories, and constructs a connective spectrum tmf_0(3).

Contribution

It develops a functorial framework for Tmf with level structures using logarithmic geometry and constructs a sheaf of elliptic cohomology theories on the moduli of elliptic curves.

Findings

Constructed a sheaf of elliptic cohomology theories with multiplicative structure.

Produced a natural restriction to cusps, linking Tmf with K-theory.

Built a connective spectrum tmf_0(3) consistent with prior conjectures.

Abstract

The cohomology theory known as 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 Tmf with level structure to forms of K-theory. In particular, this allows us to construct a connective spectrum tmf_0(3) consistent with properties suggested by Mahowald and Rezk. This is accomplished using the machinery of logarithmic structures. We construct a sheaf of locally even-periodic elliptic cohomology theories, equipped with highly structured multiplication, on the log-\'etale site of the moduli of elliptic curves. Evaluating this sheaf on modular curves produces Tmf with level structure.