Topological modular forms with level structure: decompositions and duality

TL;DR

This paper studies the decomposition and duality properties of topological modular forms with level structure, revealing their additive decompositions and self-duality characteristics under certain conditions.

Contribution

It demonstrates additive decompositions of topological modular forms with level structure and identifies cases of self-Anderson duality, extending understanding of their algebraic and duality properties.

Findings

Decomposition of TMF with level structure into simple components

Identification of self-Anderson duality in certain TMF_1(n) cases

Analysis of duality with natural C_2-action

Abstract

Topological modular forms with level structure were introduced in full generality by Hill and Lawson. We show that these decompose additively in many cases into a few simple pieces and give an application to equivariant $TMF$. Furthermore, we show which $Tmf_1(n)$ are self-Anderson dual up to a shift, both with and without their natural $C_2$-action.