The Asai--Flach Euler system in $p$-adic families

TL;DR

This paper demonstrates the $p$-adic interpolation of the Asai Euler system associated with Hilbert modular forms within a Hida family, advancing the understanding of $p$-adic properties of automorphic forms and their Galois representations.

Contribution

It extends the construction of the Asai Euler system to a $p$-adic family setting, enabling new applications in Iwasawa theory and the Bloch--Kato conjecture.

Findings

Successfully interpolates the Asai Euler system in $p$-adic families.

Provides foundational tools for proving cases of the Bloch--Kato conjecture.

Enhances understanding of $p$-adic variation of automorphic Galois representations.

Abstract

We show that the Euler system for the Asai representation corresponding to a Hilbert modular eigenform over a real quadratic field, constructed by Lei, Loeffler and Zerbes (2018), can be interpolated $p$-adically as the Hilbert modular form varies in a Hida family. This work is used as an important input in recent work of Grossi, Loeffler and Zerbes (2025) on the proof of the Bloch--Kato conjecture in analytic rank zero for the Asai representation.