A newform theory for Katz modular forms

TL;DR

This paper establishes a strong multiplicity one theorem for Katz modular forms, linking eigenforms with irreducible Galois representations to unique Katz newforms, and extends results to reducible cases.

Contribution

It introduces a new multiplicity one theorem for Katz modular forms, connecting eigenforms with Galois representations to their unique newform counterparts.

Findings

Cuspidal Katz eigenforms with irreducible Galois representations are in the old space of a unique Katz newform.

Multiplicity one results are set up for Katz eigenforms with reducible Galois representations.

Abstract

In this paper, a strong multiplicity one theorem for Katz modular forms is studied. We show that a cuspidal Katz eigenform which admits an irreducible Galois representation is in the level and weight old space of a uniquely associated Katz newform. We also set up multiplicity one results for Katz eigenforms which have reducible Galois representation.