p-parity conjecture for Panchishkin Galois representations
Abstract
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 ℓ≠ 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.
Related
- On the observability of Galois representations and the Tate conjecture
- Lifting mod pn Galois representations
- Revisiting the infinite Galois correspondence via group representations
- On the modularity of residually reducible Galois representations of rank 2
- On the variation of slopes over moduli stacks of Galois representations
BibTeX
@misc{garai2026pparity,
title = {{$p$-parity conjecture for Panchishkin Galois representations}},
author = {Subham Garai and Somnath Jha and Sudhanshu Shekhar},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08316},
eprint = {2611.08316},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08316},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - p-parity conjecture for Panchishkin Galois representations
AU - Subham Garai
AU - Somnath Jha
AU - Sudhanshu Shekhar
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08316
UR - https://arxiv.org/abs/2611.08316
AB - Let p be an odd prime, and let V_1 and V_2 be symplectic self-dual p-adic de Rham representations of G_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 ℤ_p-extension: a new invariant introduced at p, and local Iwasawa λ-invariants at ℓ≠ 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 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.
ER -