Topological modular forms with level structure

Michael Hill, Tyler Lawson

Research output: Contribution to journalArticlepeer-review

13 Scopus citations

Abstract

The cohomology theory known as (Formula presented.), 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 (Formula presented.) with level structure to forms of (Formula presented.)-theory. In particular, this allows us to construct a connective spectrum (Formula presented.) 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 (Formula presented.) with level structure.

Original languageEnglish (US)
Pages (from-to)359-416
Number of pages58
JournalInventiones Mathematicae
Volume203
Issue number2
DOIs
StatePublished - Feb 1 2016

Bibliographical note

Funding Information:
M. Hill was partially supported by NSF DMS–0906285, DARPA FA9550–07–1–0555, and the Sloan foundation. Tyler Lawson was partially supported by NSF DMS–1206008 and the Sloan foundation.

Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.

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