Parity-induced Selmer growth for symplectic, ordinary families

Let p be an odd prime, and let K/K0 be a quadratic extension of number fields. Denote by K± the maximal Zp-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 of proof generalizes a method of Mazur and 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.