Finding large Selmer rank via an arithmetic theory of local constants
成果类型：
Article
署名作者：
Mazur, Barry; Rubin, Karl
刊物名称：
ANNALS OF MATHEMATICS
ISSN/ISSBN：
0003-486X
DOI：
10.4007/annals.2007.166.579
发表日期：
2007
页码：
579-612
关键词：
摘要：
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 Z(p)-rank of the pro-p Selmer group S-p(E/K) is odd, then rank(Zp)S(p)(E/F) >= [F:K] for every finite extension F of K in K-.