$p$-parity conjecture for Panchishkin Galois representations

Abstract

Let $p$ be an odd prime, and let $V_1$ and $V_2$ be symplectic self-dual $p$-adic de Rham representations of $G_{\mathbb{Q}}$ that are pure (of weight $-1$) and satisfy the Panchishkin condition at $p$. Suppose that $V_1$ and $V_2$ are residually isomorphic, compatibly with their Panchishkin filtrations. We prove the relative $p$-parity conjecture relating the ratio of the global root numbers of $V_1$ and $V_2$ to the difference of their Bloch-Kato Selmer dimensions. We have no restriction on the dimension of the Galois representation, the Hodge-Tate weights or the monodromy operator at $p$, and the residual isomorphism need not be symplectic. The proof expresses ratios of local root numbers in terms of local invariants over the cyclotomic $\mathbb{Z}_p$-extension: a new invariant introduced at $p$, and local Iwasawa $λ$-invariants at $\ell\neq p$. Both are compatible with the arithmetic local constants of Mazur-Rubin and Nekovář. A key input, proved using Sen theory, is that the Bloch-Kato and strict local conditions at $p$ coincide over the cyclotomic $\mathbb{Z}_p$-extension, generalising results of Coates-Greenberg and Nekovář. As applications, we obtain new cases of the $p$-parity conjecture, including cases of rank greater than two: Artin twists of CM Hida families, orthogonal Artin twists of ordinary modular forms, symmetric cubes along Hida families, and families of abelian surfaces, together with unconditional numerical examples.