The growth of the rank of Abelian varieties upon extensions

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We study the growth of the rank of elliptic curves and, more generally, Abelian varieties upon extensions of number fields. First, we show that if L/K is a finite Galois extension of number fields such that (L/K) does not have an index 2 subgroup and A/K is an Abelian variety, then A(L)- A(K) can never be 1. We obtain more precise results when (L/K) is of odd order, alternating, ₂(ₚ) or ₂(ₚ). This implies a restriction on E(K(E[p]))- E(K(ζₚ)) when E/K is an elliptic curve whose mod p Galois representation is surjective. Similar results are obtained for the growth of the rank in certain non-Galois extensions. Second, we show that for every n≥2 there exists an elliptic curve E over a number field K such that ⊗ₖ/ E contains a number field of degree 2ⁿ. We ask whether every elliptic curve E/K has infinite rank over K(2), where (2) is the compositum of all quadratic extensions of. We show that if the answer is yes, then for any n≥2, there exists an elliptic curve E/K admitting infinitely many quadratic twists whose rank is a positive multiple of 2ⁿ.