The growth of the rank of Abelian varieties upon extensions

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 $\Gal(L/K)$ does not have an index 2 subgroup and $A/K$ is an Abelian variety, then $\rk A(L)-\rk A(K)$ can never be 1. We obtain more precise results when $\Gal(L/K)$ is of odd order, alternating, $\SL_2(\F_p)$ or $\PSL_2(\F_p)$. This implies a restriction on $\rk E(K(E[p]))-\rk E(K(\zeta_p))$ 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 $n\ge2$ there exists an elliptic curve $E$ over a number field $K$ such that $\Q\otimes_\Q\Res_{K/\Q} E$ contains a number field of degree $2^n$. We ask whether every elliptic curve $E/K$ has infinite rank over $K\Q(2)$, where $\Q(2)$ is the compositum of all quadratic extensions of $\Q$. We show that if the answer is yes, then for any $n\ge2$, there exists an elliptic curve $E/K$ admitting infinitely many quadratic twists whose rank is a positive multiple of $2^n$.