Automorphic Forms and Arithmetic

In 1963, Tate conjectured that the algebraic cycles of a variety can be described in terms of the Galois invariants of certain étale cohomology groups. In this talk, I'll discuss a proof of several new cases of the Tate conjecture for abelian varieties over finite fields, extending previous results of Dupuy--Kedlaya--Zureick-Brown, Lenstra--Zarhin, Tankeev, and Zarhin. This is joint work with Santiago Arango-Piñeros and Sam Frengley.

The theory of Euler systems, first developed by Thaine and Kolyvagin, has become a central tool for proving cases of the Birch–Swinnerton-Dyer and Bloch–Kato conjectures. Many of the known examples are inspired from automorphic period integrals that capture special values of L-functions. In this talk, I will focus on recent developments for Hilbert modular forms, their p-adic L-functions and some arithmetic consequences.

The construction of Euler systems often involves delicate choices of local test vectors. Using the relative Satake isomorphism, in particular the unramified Plancherel formula, we give a conceptual proof of their existence in many settings. As an example, we will treat the Gan-Gross-Prasad and Friedberg-Jacquet case uniformly. This is joint work with Li Cai and Yangyu Fan.

In a series of four papers from 1971 to 1980, Stark proposed conjectures relating algebraic units to the leading coefficients of Artin L-functions. He also suggested an approach to Hilbert's 12th problem and explicit class field theory based on these ideas. This talk will recall the well-established theory for abelian extensions of the field of rational numbers and imaginary quadratic fields before turning towards cyclotomic extensions of complex cubic fields.

The Bernstein center of a $p$-adic group is a commutative ring of certain distributions on the group, and it interacts closely with the group’s representation theory. Fargues and Scholze provide an abstract construction of a class of elements of the Bernstein center called excursion operators, which encode a candidate for the (semisimplified) local Langlands correspondence. In this talk, I will present an approach to understanding excursion operators concretely as distributions on the group, with a special emphasis on the case of $G = \mathrm{SL}_2$ where everything can be made quite explicit.