Theta characteristics and noncongruence modular forms

TL;DR

This paper explores the relationship between theta characteristics on modular curves and noncongruence modular forms, revealing new connections and raising questions in Brill--Noether theory.

Contribution

It demonstrates that sections of certain theta characteristics produce noncongruence modular forms and introduces a twisted period map related to the moduli of abelian surfaces.

Findings

Sections of theta characteristics yield noncongruence modular forms.

A twisted period map arises from non-equal theta characteristics.

Questions on Brill--Noether theory of modular curves are addressed.

Abstract

The Hodge bundle $\omega$ over a modular curve is a square-root of the canonical bundle twisted by the cuspidal divisor, or a theta characteristic, due to the Kodaira--Spencer isomorphism. We prove that, in most cases, a section of a theta characteristic $\nu$ (or any odd power of it) different from $\omega$ is a noncongruence modular form. On the other hand, we show how $\nu\ne\omega$ gives rise to a ``twisted'' analogue of the diagonal period map to a Siegel threefold, whose difference attributes to the stackiness of the moduli of abelian surfaces $\mathcal{A}_{2}$. Some questions on the Brill--Noether theory of the modular curves are answered.