New perspectives on $p$-adic regulator formulae

TL;DR

The paper develops a streamlined method to prove $p$-adic regulator formulas for Asai--Flach and diagonal classes in finite slope settings, avoiding finite polynomial cohomology. It achieves this via derived de Rham comparison, homological algebra, and a pullback/reconstruction framework (inspired by SV–Skinner) that relocates computations to ordinary or dagger loci. The main results relate Bloch–Kato log maps of regulator classes to central values of $p$-adic $L$-functions, providing explicit formulas and extending to twisted triple products and related diagonal constructions. The approach yields a flexible, ramification-tolerant toolkit for interpolating $p$-adic regulators and L-values across families of automorphic forms. The techniques hinge on Clebsch–Gordan pullbacks, dagger-analytic methods, and coherent/rigid cohomology, offering practical simplifications for regulator computations in non-ordinary or finite slope contexts.

Abstract

We generalise the proof of the $p$-adic regulator formula for Asai--Flach classes to the finite slope case, without using finite polynomial cohomology. Moreover, we simplify the analogous computation for diagonal classes, relying on a pullback construction inspired by recent work of Sangiovanni-Vincentelli--Skinner.

Theorems & Definitions (74)

• Lemma 2.1
• proof
• Lemma 2.2
• Remark 2.3
• proof : Proof of Lemma \ref{['L202']}
• Remark 3.1
• Lemma 3.2: SV-S
• Remark 3.3
• Proposition 3.4
• Remark 3.5