On integral aspects of Asai periods and Euler systems for $\mathrm{Res}_{E/\mathbf{Q}}\mathrm{GL}_2$

TL;DR

This paper investigates the integral properties of Asai periods and Euler systems for Hilbert modular forms over totally real quadratic fields, using harmonic analysis and representation theory to prove conjectures and extend known results.

Contribution

It introduces a new harmonic analysis approach to study integral Asai periods and proves conjectured integral behaviors of local factors in Euler systems, extending previous results.

Findings

Proved the integral behavior of local factors in tame norm relations.

Established the most general version of Asai-Flach Euler system tame norm relations.

Extended results of Grossi to broader integral collections.

Abstract

Let $E/\mathbf{Q}$ be a totally real quadratic field. Using unramified harmonic analysis in Hecke modules, we study the $\ell$-adic integral behavior of the (unramified part of the) Asai period attached to a Hilbert modular form for $E$, when evaluated on arbitrary integral test data in the sense of Loeffler. Using the same representation-theoretic framework, we also prove the conjectured integral behavior of local factors appearing in tame norm relations, between any collection of integral motivic Asai-Flach classes in the recipe of Loeffler-Skinner-Zerbes. Finally, specializing to one such specific integral collection, we obtain the most general version of the Asai-Flach Euler system tame norm relations, extending a result of Grossi.