Rankin--Eisenstein classes in Coleman families

TL;DR

This paper demonstrates that the Euler system linked to Rankin--Selberg convolutions of modular forms varies analytically within p-adic Coleman families, establishing an explicit reciprocity law and advancing cases of the Bloch--Kato conjecture.

Contribution

It introduces the variation of the Euler system in Coleman families and proves an explicit reciprocity law, connecting to the Bloch--Kato conjecture.

Findings

Euler system varies analytically in Coleman families

Established an explicit reciprocity law for these families

Proved new cases of the Bloch--Kato conjecture

Abstract

We show that the Euler system associated to Rankin--Selberg convolutions of modular forms, introduced in our earlier works with Lei and Kings, varies analytically as the modular forms vary in $p$-adic Coleman families. We prove an explicit reciprocity law for these families, and use this to prove cases of the Bloch--Kato conjecture for Rankin--Selberg convolutions.