An anticyclotomic Euler system for adjoint modular Galois representations

TL;DR

This paper constructs an anticyclotomic Euler system for adjoint Galois representations associated with modular forms over imaginary quadratic fields, linking it to p-adic L-functions and advancing Bloch-Kato and Iwasawa conjectures.

Contribution

It introduces a new anticyclotomic Euler system for adjoint representations and connects it to p-adic L-functions, enabling progress on Bloch-Kato and Iwasawa main conjectures.

Findings

Constructed an anticyclotomic Euler system for adjoint Galois representations.

Established relations between the Euler system and p-adic L-functions.

Derived new cases of the Bloch-Kato conjecture and divisibility results for Iwasawa theory.

Abstract

Let $K$ be an imaginary quadratic field and $p$ a prime split in $K$. In this paper we construct an anticyclotomic Euler system for the adjoint representation attached to elliptic modular forms base changed to $K$. We also relate our Euler system to a $p$-adic $L$-function deduced from the construction by Eischen-Wan and Eischen-Harris-Li-Skinner of $p$-adic $L$-functions for unitary groups. This allows us to derive new cases of the Bloch-Kato conjecture in rank zero, and a divisibility towards an Iwasawa main conjecture.