Existential definability and diophantine stability

Let K be a number field, let L be an algebraic (possibly infinite degree) extension of K, and let OK & SUB; OL be their rings of integers. Suppose A is an abelian variety defined over K such that A(K) is infinite and A(L)/A(K) is a torsion group. If at least one of the following conditions is satisfied:1. L is a number field,2. L is totally real,3. L is a quadratic extension of a totally real field,then OK has a diophantine definition over OL.& COPY; 2023 Elsevier Inc. All rights reserved.