Automorphic Periods for $\mathrm{GL}(2)$ over Totally Real Fields: Euler Systems, $p$-adic $L$-functions, and applications

Date
2025/06/13 Fri 13:30 - 14:30
Room
3号館552号室
Speaker
Giada Grossi
Affiliation
Sorbonne Paris North University
Abstract

The theory of Euler systems, initiated by the foundational works of Thaine and Kolyvagin in the late 1980s, has become a powerful tool in addressing notable cases of the Birch–Swinnerton-Dyer and Bloch–Kato conjectures. The known constructions typically originate from period integrals connected to special values of $L$-functions.

In this talk, I will focus on automorphic representations that appear in the cohomology of Hilbert modular varieties. Specifically, a variant of the Rankin–Selberg integral yields, on the one hand, the Asai–Flach Euler system and, on the other, a $p$-adic Asai $L$-function. I will explain how the connection between these two—established in joint work with D. Loeffler and S. Zerbes—leads to new cases of the Bloch–Kato conjecture for the Asai motive associated with a quadratic Hilbert modular form.

In parallel, I will discuss ongoing work with A. Graham on the twisted triple product $L$-function. In this case, the period integral appearing in Ichino’s formula plays a key role in the construction of a $p$-adic $L$-function, with the ultimate goal of tackling higher-rank cases of a twisted Birch–Swinnerton-Dyer conjecture.