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VOL. 86 | 2020 Syntomic regulators of Asai–Flach classes
David Loeffler, Christopher Skinner, Sarah Livia Zerbes

Editor(s) Masato Kurihara, Kenichi Bannai, Tadashi Ochiai, Takeshi Tsuji

Abstract

In this paper, we derive a formula for the $p$-adic syntomic regulators of Asai–Flach classes. These are cohomology classes forming an Euler system associated to a Hilbert modular form over a quadratic field, introduced in an earlier paper [LLZ16] by Antonio Lei and the first and third authors. The formula we develop here is expressed in terms of differential operators acting on overconvergent Hilbert modular forms; it is analogous to existing formulae for the regulators of Beilinson–Flach classes, but a novel feature is the appearance of a projection operator associated to a critical-slope Eisenstein series. We conclude the paper with numerical calculations giving strong evidence for the non-vanishing of these regulators in an explicit example.

Information

Published: 1 January 2020
First available in Project Euclid: 12 January 2021

Digital Object Identifier: 10.2969/aspm/08610595

Subjects:
Primary: 11F41 , 11F67 , 11F80 , 19F27

Keywords: Asai $L$-functions , Hilbert modular forms , regulators , syntomic cohomology

Rights: Copyright © 2020 Mathematical Society of Japan

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