p-parity conjecture for Panchishkin Galois representations

Let p be an odd prime, and let V₁ and V₂ be symplectic self-dual p-adic de Rham representations of GQ that are pure (of weight $-1$) and satisfy the Panchishkin condition at p. Suppose that V₁ and V₂ are residually isomorphic, compatibly with their Panchishkin filtrations. We prove the relative p-parity conjecture relating the ratio of the global root numbers of V₁ and V₂ 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 Zₚ-extension: a new invariant introduced at p, and local Iwasawa $λ$-invariants at ≠ 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 Zₚ-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.