Strictly Commutative Complex Orientations of $Tmf_1(N)$

TL;DR

This paper develops strictly commutative complex orientations for spectra of topological modular forms with level structure, extending previous methods and results in the field.

Contribution

It adapts and combines existing methods to construct strictly commutative complex orientations for $Tmf_1(N)$ spectra, advancing the understanding of their algebraic and topological properties.

Findings

Constructed strictly commutative complex orientations for $Tmf_1(N)$.

Extended methods of Ando, Hopkins, Rezk, Wilson, Lawson.

Enhanced the theoretical framework for topological modular forms with level structures.

Abstract

We construct by adapting methods and results of Ando, Hopkins, Rezkand Wilson combined with results of Hopkins and Lawson strictly commu-tative complex orientations for the spectra of topological modular forms with level $\Gamma_1(N)$.