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.