The Failure of Galois Descent for p-Selmer Groups of Elliptic Curves

We show that if F is Q or a multiquadratic number field, p∈\2,3,5\ , and $K/F$ is a Galois extension of degree a power of p , then for elliptic curves E/Q ordered by height, the average dimension of the p -Selmer groups of $E/K$ is bounded. In particular, this provides a bound for the average K -rank of elliptic curves E/Q for such K . Additionally, we give bounds for certain representation–theoretic invariants of Mordell–Weil groups over Galois extensions of such F . The central result is that: for each finite Galois extension $K/F$ of number fields and prime number p , as E/Q varies, the difference in dimension between the Galois fixed space in the p -Selmer group of $E/K$ and the p -Selmer group of $E/F$ has bounded average.