Generic expansion of an abelian variety by a subgroup

TL;DR

This paper proves that expanding an abelian variety with a generic divisible subgroup yields a theory with specific model-theoretic properties, under the condition that the variety has endomorphism ring equal to integers.

Contribution

It establishes the existence of such expansions for abelian varieties with endomorphism ring $ ext{End}(A)= extbf{Z}$ and explores extensions to simple abelian varieties.

Findings

The expanded theory is $ ext{NSOP}_1$ and not simple.

Existence of abelian varieties with $ ext{End}(A)= extbf{Z}$ for any genus.

Extension to simple abelian varieties via submodule predicates.

Abstract

Let $A$ be an abelian variety in a field of characteristic $0$. We prove that the expansion of $A$ by a generic divisible subgroup of $A$ with the same torsion exists provided $A$ has few algebraic endomorphisms, namely $\mathrm{End}(A)=\mathbb Z$. The resulting theory is $\mathrm{NSOP}_1$ and not simple. Note that there exist abelian varieties $A$ with $\mathrm{End}(A) = \mathbb{Z}$ of any genus. We indicate how this result can be extended to any simple abelian variety by considering the expansion by a predicate for some submodule over $\mathrm{End}(A)$.