Diagonal cycles and anticyclotomic twists of modular forms at inert primes

TL;DR

This paper constructs an anticyclotomic Euler system for modular forms using diagonal classes and applies it to obtain new results related to the Bloch--Kato conjecture at inert primes.

Contribution

It introduces a novel construction of an anticyclotomic Euler system via diagonal classes and advances understanding of the Bloch--Kato conjecture in rank one cases.

Findings

New results towards the Bloch--Kato conjecture at inert primes.

Construction of an anticyclotomic Euler system using diagonal classes.

Application of previous results to analytic rank one scenarios.

Abstract

We revisit the construction of Castella and Do of an anticyclotomic Euler system for the $p$-adic Galois representation of a modular form, using diagonal classes. Combining this construction and some previous results of ours, we obtain new results towards the Bloch--Kato conjecture in analytic rank one, assuming that the fixed prime $p$ is inert in the relevant imaginary quadratic field.