Topological modular forms with level structure

TLDR: The authors of as discussed by the authors constructed a presheaf of locally even-periodic elliptic cohomology theories, equipped with highly structured multiplication, on the log-etale site of the moduli of elliptic curves.

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-etale site of the moduli of elliptic curves. Evaluating this presheaf on modular curves produces $$\mathrm{Tmf}$$ with level structure.