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 (., 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.
