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

TL;DR

The paper develops a comprehensive integral framework for Asai periods attached to Hilbert modular forms over a real quadratic field, proving $"l$-adic integrality$ of unramified Asai periods and linking these to the integrality of local factors in tame norm relations. It extends the Rankin–Selberg–type integral approach to the Asai setting, establishing a freeness/finiteness structure for local Hecke modules and constructing explicit integral test data that control local periods via Satake parameters and Asai $L$-factors. A central achievement is the integral version of the Asai-Flach norm relations for motivic classes, showing divisibility by the Asai Euler factors modulo $p-1$ and providing a most general formulation across all unramified primes, including a cyclotomic base-change perspective through the smaller group $oldsymbol{rak G}^*$. These results yield a robust integral Euler-system framework for Hilbert modular forms and set the stage for applications in $ ext{p}$-adic $L$-functions and Iwasawa theory, with natural extensions to nontrivial coefficient systems. The methods blend local harmonic analysis, integral lattices in Hecke modules, and motivic cohomology, offering a unified approach to the integrality phenomena predicted for Asai periods and their associated Euler systems.

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.

Theorems & Definitions (48)

• Theorem A: \ref{['thm hilbert period']}
• Theorem B: \ref{['thm euler system']}
• Definition 2.0.1
• Theorem 3.1.1
• proof
• Corollary 3.1.2
• proof
• Proposition 3.2.1
• proof
• Proposition 3.2.2