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 $K\mathrm{/}k$ is a quadratic extension of number fields, $E$ is an elliptic curve defined over $k$, and $p$ is an odd prime. Let $F$ 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 $Gal(K\mathrm{/}k)$ acts on $Gal(F\mathrm{/}K)$ by -1). We prove (under mild hypotheses on $p$) that if the rank of the pro-$p$ Selmer group $S_p(E\mathrm{/}K)$ is odd, then the rank of $S_p(E\mathrm{/}L)$ is at least $[L:K]$ for every finite extension $L$ of $K$ in $F$.