# Finding large Selmer rank via an arithmetic theory of local constants | Annals of Mathematics

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 − 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) by inversion). We prove (under mild hypotheses on p) that if the Zp-rank of the pro-p Selmer group p(E/K) is odd, then rankZpp(E/F)≥[F:K] for every finite extension F of K in −.

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### 数学年刊（Annals of Mathematics）

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