Geometric and $p$-adic modular forms of half-integral weight

TL;DR

This paper develops a geometric framework for half-integral weight modular forms, introduces geometric Hecke operators, and explores initial properties of $p$-adic forms, laying groundwork for future research.

Contribution

It presents a novel geometric formalism for half-integral weight modular forms and constructs geometric Hecke operators, advancing the understanding of their structure.

Findings

Construction of geometric Hecke operators

Introduction of basic $p$-adic form spaces

Foundational properties of the geometric formalism

Abstract

We introduce a geometric formalism for studying modular forms of half-integral weight and explore some of its basic properties. Geometric Hecke operators are constructed and some basic spaces of $p$-adic forms are introduced. The $p$-adic theory is greatly expanded in subsequent papers, making that part of this paper largely obsolete.