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 \(K−\) 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 \(S_p(E∕K)\) is odd, then \(rankZ_p S_p(E∕F) \ge [F : K]\) for every finite extension \(F\) of \(K\) in \(K−\).