A difference formula of p-adic height pairings via the Bloch-Kato logarithm map
Abstract
The construction of a p-adic height pairing for a geometric p-adic representation of the absolute Galois group of a number field depends on a global p-adic logarithm and on local splittings of the Hodge filtrations at the primes above p. We study the dependence on these splittings for suitable two-dimensional symplectic self-dual representations, including self-dual twists of representations attached to even-weight newforms at non-ordinary primes not dividing the level. We express the difference between the height pairings associated with the two splittings determined by Frobenius explicitly in terms of local Bloch--Kato logarithms. As an application over ℚ, we prove that at least one of the two cyclotomic p-adic height pairings is non-trivial under the additional assumptions that the Frobenius eigenvalues at p are distinct and the localization map at p from the Bloch--Kato Selmer group is non-zero.
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BibTeX
@misc{adachi2026difference,
title = {{A difference formula of $p$-adic height pairings via the Bloch-Kato logarithm map}},
author = {Taiga Adachi and Yu Katagiri and Ryota Shii},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35626},
eprint = {2609.35626},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35626},
}
RIS
TY - UNPB
TI - A difference formula of p-adic height pairings via the Bloch-Kato logarithm map
AU - Taiga Adachi
AU - Yu Katagiri
AU - Ryota Shii
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35626
UR - https://arxiv.org/abs/2609.35626
AB - The construction of a p-adic height pairing for a geometric p-adic representation of the absolute Galois group of a number field depends on a global p-adic logarithm and on local splittings of the Hodge filtrations at the primes above p. We study the dependence on these splittings for suitable two-dimensional symplectic self-dual representations, including self-dual twists of representations attached to even-weight newforms at non-ordinary primes not dividing the level. We express the difference between the height pairings associated with the two splittings determined by Frobenius explicitly in terms of local Bloch--Kato logarithms. As an application over ℚ, we prove that at least one of the two cyclotomic p-adic height pairings is non-trivial under the additional assumptions that the Frobenius eigenvalues at p are distinct and the localization map at p from the Bloch--Kato Selmer group is non-zero.
ER -