Modular abelian varieties over number fields

TL;DR

This paper characterizes certain abelian varieties over Galois number fields whose zeta functions factor into products of modular forms, introducing the concept of strongly modular varieties and exploring their properties and examples.

Contribution

It provides a new characterization of strongly modular abelian varieties over Galois number fields, linking their structure to Galois cohomology and isogenies, and studies specific examples with quaternionic multiplication.

Findings

Characterization of strongly modular abelian varieties over Galois fields.

Proof of strong modularity for specific abelian surfaces with quaternionic multiplication.

Methods to construct nontrivial examples of strongly modular varieties.

Abstract

The main result of this paper is a characterization of the abelian varieties $B/K$ defined over Galois number fields with the property that the zeta function $L(B/K;s)$ is equivalent to the product of zeta functions of non-CM newforms for congruence subgroups $\Gamma_1(N)$. The characterization involves the structure of End(B), isogenies between the Galois conjugates of $B$, and a Galois cohomology class attached to $B/K$. We call the varieties having this property strongly modular. The last section is devoted to the study of a family of abelian surfaces with quaternionic multiplication. As an illustration of the ways in which the general results of the paper can be applied we prove the strong modularity of some particular abelian surfaces belonging to that family, we show how to find nontrivial examples of strongly modular varieties by twisting, and prove the existence of strongly modular surfaces satisfying certain properties.