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

Abstract 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.