Research

My research topic mainly concerns mod p geometry of Shimura varieties and level raising/lowering of automorphic forms. I am also interested in categorical Langlands.

Lever lowering for U(1,2)

Mazur’s principle provides simple conditions so that an irreducible unramified $\mathbb{F}\ell$ representation coming from a modular form of level $\Gamma{0} (Np)$ also comes from a form of level $\Gamma_0 (N)$. We prove Mazur’s principle for $U(1,2)$ by studying the geometry of special fiber of Shimura varieties.

Arithmetic level raising for U(2r,1) at an inert prime

Arithmetic level raising concerns congruences between automorphic representations of different levels at a prime, playing a crucial role in the construction of Euler systems and the Beilinson–Bloch–Kato conjecture. In this talk, we establish arithmetic level raising for the unitary group U(1,2r) at an inert prime by analyzing the mod p geometry of the corresponding Shimura variety. This case is more intricate than that of U(1,2r−1) at an inert prime, which was previously treated by Liu, Tian, and Xiao. The key ingredients include a variant of Ihara’s lemma and the observation that the map between the special fibers of Shimura varieties with certain level structures is semi-small. This semi-smallness enables us to determine the monodromy filtration on the nearby cycles of a Steinberg-type local system, which, via Ihara’s lemma, is closely related to the level raising map. This is joint work with Ruiqi Bai.