Parity-induced Selmer Growth For Symplectic, Ordinary Families

Let $p$ be an odd prime, and let $K\mathrm{/}{K}_0$ be a quadratic extension of number fields. Denote by $K_{\pm }$ the maximal ${\mathbb{Z}}_p$-power extensions of $K$ that are Galois over $K_0$, with $K_{+}$ abelian over $K_0$ and $K_{-}$ dihedral over $K_0$. In this paper we show that for a Galois representation over $K_0$ satisfying certain hypotheses, if it has odd Selmer rank over $K$ then for one of $K_{\pm }$ its Selmer rank over $L$ is bounded below by $[L:K]$ for $L$ ranging over the finite subextensions of $K$ in $K_{\pm }$. Our method or proof generalizes a method of Mazur–Rubin, building upon results of Nekov'a\v{r}, and applies to abelian varieties of arbitrary dimension, (self-dual twists of) modular forms of even weight, and (twisted) Hida families.