Galois families of modular forms and application to weight one

TL;DR

This paper introduces Galois families of modular forms derived from Galois representations over rational function fields, providing examples and constructing an infinite family of non-liftable weight one Katz modular eigenforms over finite fields.

Contribution

It defines Galois families of modular forms and constructs an infinite family of non-liftable weight one Katz modular eigenforms, expanding understanding of modular forms in positive characteristic.

Findings

Existence of Galois families of modular forms.

Construction of an infinite Galois family of non-liftable weight one forms.

Examples over algebraic closures of finite fields.

Abstract

We introduce Galois families of modular forms. They are a new kind of family coming from Galois representations of the absolute Galois groups of rational function fields over the rational field. We exhibit some examples and provide an infinite Galois family of non-liftable weight one Katz modular eigenforms over an algebraic closure of F_p for p in {3,5,7,11}.