Rankin--Eisenstein classes for modular forms

TL;DR

This paper systematically studies Rankin-Eisenstein motivic cohomology classes linked to modular forms, computing their p-adic regulators and confirming cases of the Perrin-Riou conjecture for Rankin--Selberg convolutions.

Contribution

It provides the first explicit computation of p-adic syntomic regulators for these classes and verifies the Perrin-Riou conjecture in new cases.

Findings

Computed p-adic syntomic regulators of Rankin-Eisenstein classes

Proved many cases of the Perrin-Riou conjecture for Rankin--Selberg convolutions

Established a systematic framework for motivic cohomology classes of modular forms

Abstract

In this paper we make a systematic study of certain motivic cohomology classes ("Rankin-Eisenstein classes") attached to the Rankin--Selberg convolution of two modular forms of weight $\ge 2$. The main result is the computation of the $p$-adic syntomic regulators of these classes. As a consequence we prove many cases of the Perrin-Riou conjecture for Rankin--Selberg convolutions of cusp forms.