The modularity of an abelian variety

TL;DR

This paper explores the concept of modularity for abelian varieties over the rationals, extending ideas from elliptic curves, and conjectures that all simple abelian varieties over Q are modular.

Contribution

It introduces the notion of modularity for abelian varieties and extends the concept from elliptic curves to higher-dimensional cases.

Findings

Proposes the concept of modularity for abelian varieties.

Discusses the modularity over the rational number field.

Conjectures that all simple abelian varieties over Q are modular.

Abstract

We introduce the concept of the modularity of an abelian variety defined over the rational number field extending the modularity of an elliptic curve. We discuss the modularity of an abelian variety over the rational number field. We conjecture that a simple abelian variety over the rational number field is modular.