Let be an odd prime number, let be an elliptic curve over a number field , and let be a Galois extension of degree twice a power of . We study the -corank of the -power Selmer group of over . We obtain lower bounds for , generalizing the results in [MR], which applied to dihedral extensions.
If is the (unique) quadratic extension of in , if , if is the subgroup of elements of commuting with a choice of involution of over , and if is odd, then we show that (under mild hypotheses) .
As a very specific example of this, suppose that is an elliptic curve over with a rational torsion point of order and without complex multiplication. If is an elliptic curve over with good ordinary reduction at such that every prime where both and have bad reduction has odd order in and such that the negative of the conductor of is not a square modulo , then there is a positive constant depending on but not on or such that for every
"Growth of Selmer rank in nonabelian extensions of number fields." Duke Math. J. 143 (3) 437 - 461, 15 June 2008. https://doi.org/10.1215/00127094-2008-025