Syntomic regulators of Asai--Flach classes

TL;DR

This paper derives a formula for p-adic syntomic regulators of Asai--Flach classes linked to Hilbert modular forms, introducing a novel projection operator and providing numerical evidence of non-vanishing in specific cases.

Contribution

It presents a new explicit formula for p-adic syntomic regulators of Asai--Flach classes using differential operators and a novel projection operator, extending existing regulator formulas.

Findings

Derived a formula for p-adic syntomic regulators involving differential operators.

Introduced a projection operator related to a critical-slope Eisenstein series.

Numerical calculations support non-vanishing of regulators in an example.

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 (arXiv:1607.07813) 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.