  Let p be an odd prime number, E an elliptic curve over a number field k, and
F/k a Galois extension of degree twice a power of p. We study the Z_p-corank
rk_p(E/F) of the p-power Selmer group of E over F. We obtain lower bounds for
rk_p(E/F), generalizing the results in [MR], which applied to dihedral
extensions.
  If K is the (unique) quadratic extension of k in F, G = Gal(F/K), G^+ is the
subgroup of elements of G commuting with a choice of involution of F over k,
and rk_p(E/K) is odd, then we show that (under mild hypotheses) rk_p(E/F) \ge
[G:G^+]$.
  As a very specific example of this, suppose A is an elliptic curve over Q
with a rational torsion point of order p, and with no 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 F_p^\times,
and such that the negative of the conductor of E is not a square mod p, then
there is a positive constant B, depending on A but not on E or n, such that
rk_p(E/Q(A[p^n])) \ge B p^{2n} for every n.
