Euler systems for Rankin--Selberg convolutions of modular forms

TL;DR

This paper constructs an Euler system for tensor products of Galois representations from modular forms, using higher Chow groups, and applies it to prove finiteness results for Selmer groups under certain conditions.

Contribution

It introduces a new Euler system in the cohomology of tensor products of Galois representations from modular forms, linking algebraic cycles to Selmer group finiteness.

Findings

Euler system constructed in the cohomology of tensor product Galois representations.

Finiteness of the strict Selmer group proved when p-adic L-function is non-zero at s=1.

Connection established between higher Chow groups and Selmer group properties.

Abstract

We construct an Euler system in the cohomology of the tensor product of the Galois representations attached to two modular forms, using elements in the higher Chow groups of products of modular curves. We use this Euler system to prove a finiteness theorem for the strict Selmer group of the Galois representation when the associated p-adic Rankin--Selberg L-function is non-vanishing at s = 1.