Fields of definition of abelian subvarieties

TL;DR

This paper investigates the fields over which abelian subvarieties are defined, showing conditions for infinitely many such subvarieties to share a common field of definition, and providing bounds on minimal extension degrees.

Contribution

It establishes the existence of infinitely many abelian subvarieties with a specific field of definition under certain conditions and refines bounds on minimal extension degrees for their definition.

Findings

Infinitely many abelian subvarieties share the field of definition $K_A$.

Explicit maximum for minimal degree of extension for defining subvarieties.

Results depend on the structure of isotypic components of the abelian variety.

Abstract

In this paper we study the field of definition of abelian subvarieties $B\subset A_{\overline{K}}$ for an abelian variety $A$ over a field $K$ of characteristic $0$. We show that, provided that no isotypic component of $A_{\overline{K}}$ is simple, there are infinitely many abelian subvarieties of $A_{\overline{K}}$ with field of definition $K_A$, the field of definition of the endomorphisms of $A_{\overline{K}}$. This result combined with earlier work of R\'emond gives an explicit maximum for the minimal degree of a field extension over which an abelian subvariety of $A_{\overline{K}}$ is defined with varying $A$ of fixed dimension and $K$ of characteristic $0$.