Syntomic regulators of Asai--Flach classes

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.

Revised version: strengthened main theorem to avoid unnecessary assumptions, and added a formula for normalisation of Eisenstein functional

References in corpus (6)

• Rankin--Eisenstein classes and explicit reciprocity laws
• Rankin--Eisenstein classes for modular forms
• Euler systems for Hilbert modular surfaces
• Spectral expansions of overconvergent modular functions
• Twisted triple product p-adic L-functions and Hirzebruch-Zagier cycles
• p-adic Abel-Jacobi map and p-adic Gross-Zagier formula for Hilbert modular forms

Cited by in corpus (4)

• Rankin--Eisenstein classes for modular forms
• Euler systems for Hilbert modular surfaces
• On the Bloch-Kato conjecture for GSp(4)
• Eisenstein degeneration of Euler systems