Submission date: 23 December 2019

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 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. We indicate how this result can be extended to any simple abelian variety by considering the expansion by a predicate for some submodule over End(A).

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Full text arXiv 1912.10911: pdf, ps.