Rankin--Eisenstein classes and explicit reciprocity laws

TL;DR

This paper constructs p-adic families of Galois cohomology classes linked to Rankin convolutions of modular forms and establishes an explicit reciprocity law connecting these classes to L-function values, with applications to Selmer groups.

Contribution

It introduces new p-adic families of Galois cohomology classes and proves an explicit reciprocity law relating them to L-values, advancing the understanding of special values and Selmer groups.

Findings

Established explicit reciprocity law for Rankin--Eisenstein classes.

Proved finiteness of Selmer groups under non-vanishing L-values.

Constructed three-variable p-adic families of Galois cohomology classes.

Abstract

We construct three-variable $p$-adic families of Galois cohomology classes attached to Rankin convolutions of modular forms, and prove an explicit reciprocity law relating these classes to critical values of L-functions. As a consequence, we prove finiteness results for the Selmer group of an elliptic curve twisted by a 2-dimensional odd irreducible Artin representation when the associated $L$-value does not vanish.