Finding large Selmer rank via an arithmetic theory of local constants

Abstract

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 $\mathcal{K}^-$ 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 $\mathrm{Gal}(K/k)$ acts on $\mathrm{Gal}(\mathcal{K}^-/K)$ by inversion). We prove (under mild hypotheses on $p$) that if the $\mathbf{Z}_p$-rank of the pro-$p$ Selmer group $\mathcal{S}_p(E/K)$ is odd, then $\mathrm{rank}_{\mathbf{Z}_p} \mathcal{S}_p(E/F) \ge [F:K]$ for every finite extension $F$ of $K$ in $\mathcal{K}^-$.