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arXiv · 2609.35626

A difference formula of $p$-adic height pairings via the Bloch-Kato logarithm map

Taiga Adachi·Yu Katagiri·Ryota Shii
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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 $\mathbb{Q}$, 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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Taiga Adachi, Yu Katagiri, Ryota Shii. 2026-09-28. A difference formula of $p$-adic height pairings via the Bloch-Kato logarithm map. https://arxiv.org/abs/2609.35626

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