An anticyclotomic Euler system of Hirzebruch--Zagier cycles I: Norm relations and $p$-adic interpolation

TL;DR

This paper constructs an anticyclotomic Euler system for the Asai Galois representation linked to p-ordinary Hilbert modular forms over real quadratic fields, with implications for Bloch--Kato and Iwasawa conjectures.

Contribution

It introduces a new Euler system based on Hirzebruch--Zagier cycles that vary in p-adic Hida families, advancing the understanding of Galois representations.

Findings

Construction of an anticyclotomic Euler system for Asai Galois representations.

Demonstration of p-adic variation of Euler system classes in Hida families.

Applications to Bloch--Kato and Iwasawa Main Conjectures.

Abstract

We construct an anticyclotomic Euler system for the Asai Galois representation associated to $p$-ordinary Hilbert modular forms over real quadratic fields. We also show that our Euler system classes vary in $p$-adic Hida families. The construction is based on the study of certain Hirzebruch--Zagier cycles obtained from modular curves of varying level diagonally emdedded into the product with a Hilbert modular surface. By Kolyvagin's methods, in the form developed by Jetchev--Nekov\'{a}\v{r}--Skinner in the anticyclotomic setting, the construction yields new applications to the Bloch--Kato conjecture and the Iwasawa Main Conjecture.