Generic expansion of an abelian variety by a subgroup

Abstract

Let A be an abelian variety in an algebraically closed 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 End(A) = Z. The resulting theory is NSOP 1 and not simple. Note that there exist abelian varieties A with End(A) = Z of any genus.

Vector spaces with a dense-codense generic submodule

“…By Theorem 3. 16 we may assume Y = ∪ j≤n Y j where each Y j is the set of realizations of a formula of the form…”

“…Both expansions of Winkler were later shown to preserve the property NSOP 1 ( [21], [22]). One can also consider the expansion of a theory by a predicate for a reduct of this theory, for instance expanding a theory of fields by an additive or multiplicative generic subgroup (see [14,16,7]).…”