Orientations and Topological Modular Forms with Level Structure

TL;DR

This paper explores the characteristic series of topological modular forms with level structures using advanced methods, providing new examples and showing how certain genera relate to ring maps and cobordism orientations.

Contribution

It introduces new characteristic series for tmf with level structures and demonstrates how the Ochanine genus and orientations descend in specific cases.

Findings

Characteristic series for tmf_0(N) are described.

Ochanine genus arises from an E-infinity ring map.

Certain tmf orientations of String cobordism descend to Spin cobordism.

Abstract

Using the methods of Ando-Hopkins-Rezk, we describe the characteristic series arising from E-infinity genera valued in topological modular forms with level structure. We give examples of such series for tmf_0(N) and show that the Ochanine genus comes from an E-infinity ring map. We also show that, away from 6, certain tmf orientations of String cobordism descend to orientations of Spin cobordism.