Title: Covers of P^1 and number fields Abstract: Let $n$ be an integer. Via the Minkowski embedding, an order $mathcal{O}$ in a degree $n$ number field can be seen as a lattice. Similarly, given a degree $n$ cover of $mathbb{P}^1$, the pushforward of the structure sheaf is an interesting rank $n$ vector bundle on $mathbb{P^1}$. […]
Title: Kloosterman sums in representation theory of GL(n, Fq) Abstract: Given a "nice" irreducible representation of GL(n, Fq), one can define a special matrix coefficient attached to it, called the Bessel function. Often Kloosterman sums show up as special values of these Bessel functions. In this talk, I will first explain my previous results regarding […]
Title: First explicit reciprocity law for unitary Friedberg—Jacquet periods Abstract: In the early 2000's, Bertolini and Darmon introduced a new technique to bound Selmer groups of elliptic curves via level raising congruences. This was the first example of what is now termed a "bipartite Euler system", and over the last decade we have seen many […]
Title: Unipotent Representations of Real Reductive Groups and Mixed Hodge Modules Abstract: Let Pi(G) denote the set of irreducible unitary representations of a semisimple Lie group G. A fundamental problem in representation theory is to describe the structure of this set. In previous joint work with Losev and Matvieievskyi, we have defined a class of […]
Title: Nonabelian Fourier kernels on SL₂ and GL₂ Abstract: I will present my collaborative work with Ngô on explicit formulas for the nonabelian Fourier kernels on SL₂ and GL₂, as conjectured by Braverman and Kazhdan. Additionally, I will discuss its connection to the orbital Hankel transform studied by Ngô.
Title: The local theta correspondence and functoriality Abstract: In a letter to Howe, Langlands conjectured that the local theta correspondence is an instance of Langlands functoriality, e.g., it should preserve L-packets. Unfortunately, this was false. As a remedy, Adams conjectured that instead of L-packets, the local theta correspondence should preserve Arthur packets. Mœglin showed that […]
Title: Relative Satake isomorphism and Euler systems Abstract: Starting from the inversion formula for the relative Satake isomorphism due to Sakellaridis, we observe a simple additional divisibility property. We will then explain how this gives a computation-free construction of Euler systems in some settings. This is joint work with Li Cai and Yangyu Fan.
Title: On the Gan-Gross-Prasad conjecture for unitary groups Abstract: The Gan-Gross-Prasad conjecture stipulates the existence of relations between the central values of certain L-functions and some automorphic periods on classical groups. It can be seen as a higher rank generalization of a famous formula of Waldspurger for toric periods on GL(2). In this talk, I […]