The diagonal cycle Euler system for ${\rm GL}_2\times{\rm GL}_2$

TL;DR

This paper constructs an anticyclotomic Euler system for the Rankin-Selberg convolution of modular forms, leading to new results on the Bloch-Kato conjecture and Iwasawa theory.

Contribution

It introduces a novel Euler system based on diagonal cycles in p-adic families, advancing the understanding of special values and Selmer groups.

Findings

Proves new cases of the Bloch-Kato conjecture in ranks zero and one.

Establishes divisibility results towards the Iwasawa main conjecture.

Develops p-adic families of diagonal cycles for L-function studies.

Abstract

We construct an anticyclotomic Euler system for the Rankin-Selberg convolution of two modular forms, using $p$-adic families of generalized Gross-Kudla-Schoen diagonal cycles. As applications of this construction, we prove new cases of the Bloch-Kato conjecture in analytic ranks zero and one, and a divisibility towards an Iwasawa main conjecture.