Growth of Selmer rank in nonabelian extensions of number fields

Abstract

Let p be an odd prime number, let E be an elliptic curve over a number field k, and let F/k be a Galois extension of degree twice a power of p. We study the Zp-corank rkp(E/F) of the p-power Selmer group of E over F. We obtain lower bounds for rkp(E/F), generalizing the results in [MR], which applied to dihedral extensions. If K is the (unique) quadratic extension of k in F, if G=Gal(F/K), if G+ is the subgroup of elements of G commuting with a choice of involution of F over k, and if rkp(E/K) is odd, then we show that (under mild hypotheses) rkp(E/F)≥[G:G+]. As a very specific example of this, suppose that A is an elliptic curve over Q with a rational torsion point of order p and without complex multiplication. If E is an elliptic curve over Q with good ordinary reduction at p such that every prime where both E and A have bad reduction has odd order in Fp× and such that the negative of the conductor of E is not a square modulo p, then there is a positive constant B depending on A but not on E or n such that rkp(E/Q(A[pn]))≥Bp2n for every n