(Topological) modular forms with level structures: decompositions and duality

Abstract

The present article studies decompositions of vector bundles on the moduli stack of elliptic curves that are pushforwards of vector bundles on moduli of elliptic curves with level structure. These imply decomposition results for rings of modular forms and also for topological modular forms. We give explicit formulas for these decompositions and also apply them to equivariant topological modular forms. Moreover, we study the dualizing sheaf on $\overline{\mathcal{M}}_1(n)$ and characterize the numbers $n$ such that $Tmf_1(n)$ is Anderson self-dual.