Title: Twists of commutative algebraic groups and a question of Silverman

Abstract: In this talk I will report on a joint work with Antigona Pajaziti (University of Luxembourg). In Arizona Winter School 2024 Joe Silverman asked a few questions concerning the Trace map from the L-rational points to the K-rational points of an abelian variety A/K and a Galois extension L/K for a number field K. This was inspired by a previous work of Mirela Ciperiani and Ekin Ozman where they answer a special case of similar questions when A=E is an elliptic curve and [L : K]=2. We answer this question in its full generality. The main ingredient in our result is the theory of twists of commutative algebraic groups developed by many people in various special cases and further developed and compiled by Barry Mazur, Karl Rubin and Alice Silverberg. I will try to discuss the most relevant constructions in this paper and use them to answer Silverman's questions.