Finding Large Selmer Rank via an Arithmetic Theory of Local Constants

We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose K/k is a quadratic extension of number fields, E is an elliptic curve defined over k, and p is an odd prime. Let K− denote the maximal abelian p-extension of K that is unramified at all primes where E has bad reduction and that is Galois over k with dihedral Galois group (i.e., the generator c of Gal(K/k) acts on Gal(K−/K) by inversion). We prove (under mild hypotheses on p) that if the Zp-rank of the pro-p Selmer group Sp(E/K) is odd, then rankZpSp(E/F ) ≥ [F : K] for every finite extension F of K in K −.