Parity-induced Selmer Growth For Symplectic, Ordinary Families

2008/05/16 by Pottharst, Jonathan
#11R20 (Secondary) #11R23 (Primary) #11R34 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.0805.2508

Abstract

Let p be an odd prime, and let K/K₀ be a quadratic extension of number fields. Denote by K_± the maximal ℤ_p-power extensions of K that are Galois over K₀, with K₊ abelian over K₀ and K_- dihedral over K₀. In this paper we show that for a Galois representation over K₀ satisfying certain hypotheses, if it has odd Selmer rank over K then for one of K_± its Selmer rank over L is bounded below by [L:K] for L ranging over the finite subextensions of K in K_±. Our method or proof generalizes a method of Mazur--Rubin, building upon results of Nekovář, and applies to abelian varieties of arbitrary dimension, (self-dual twists of) modular forms of even weight, and (twisted) Hida families.

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