Arithmetic and Representation Theory (ART) Seminar

Abstract: In this talk, we will use type theory to construct a family of depth 1/N supercuspidal representations of p-adic GL(2N, F) which we call middle supercuspidal representations. These supercuspidals may be viewed as a natural generalization of simple supercuspidal representations, i.e. those supercuspidals of minimal positive depth. Via explicit computations of twisted gamma factors, we will show that middle supercuspidal representations may be uniquely determined through twisting by quasi-characters of F* and simple supercuspidal representations of GL(N, F). Furthermore, we will pose a conjecture which refines the Local Converse Theorem for GL(n, F).

Abstract: I will introduce the study of arithmetic subgroups of isometries of hyperbolic space (of the first type) and their quotients which are finite volume hyperbolic orbifolds. These groups are constructed from quadratic forms defined over totally real number fields and the geometry of their quotients reflects the arithmetic properties of the associated forms. I will also discuss joint work with P. Murillo concerning commensurability and the length spectrum of arithmetic hyperbolic manifolds.

Abstract: I will first give a review of many people works on Loop groups, vector bundles on complex P^1 and Langlands duality for reductive groups. Then I will discuss recent progress on extending the story to a more general setting.

Abstract: Given a family g : X -> S of smooth projective algebraic varieties over a number field K, one often wants to constrain the points s in S where the fibre X_s acquires "extra" algebraic structure. A basic sort of constraint which is important in unlikely intersection theory is that of a Galois-orbit lower bound: an inequality h(s) <= poly([K(s) : K]), where h is some logarithmic Weil height and K(s) is the field of definition of s. Recent work has focused on how to use G-functions constructed from degenerations of g to produce such inequalities.

We describe some new results in the case where g is a one-parameter degeneration of surfaces, and the central role played by rigid and "adelic" geometry.

Abstract: In this talk, we will give a description of the depth-r Bernstein center of a connected reductive p-adic group  for rational depths, as an inverse limit of algebras which we call depth-r standard parahoric Hecke algebras. We will introduce depth-r restricted Langlands parameters attached to smooth irreducible representations and use our description to construct projectors and give a decomposition of the category of smooth representations into a product of full subcategories indexed by restricted Langlands parameters. This is based on a joint work with Tsao-Hsien Chen.  

Abstract: The quadratic Chabauty method is a powerful technique for computing rational points on curves. In this talk, I will discuss possible failure cases of the method, where the set of points it outputs contains extra “unexpected” irrational points. I will explain a heuristic regarding the appearance of these unexpected points, and describe some work with Jennifer Balakrishnan in which we almost completely classify unexpected points in a simple test case. In the end, this reduces to studying certain explicit polynomials in two variables, with the aid of a computer.

Abstract: The study of the geometry and arithmetic of Shimura varieties, and more generally, period domains, have had important applications in number theory and particularly in the Langlands program.  A theorem of Borel from the 1970s says that any holomorphic map from a smooth complex algebraic variety to a Shimura variety is automatically an algebraic map. In this talk, I will discuss a p-adic analogue of this result both in the context of general Shimura varieties and for period domains. In particular, I will explain some inputs from the theory of prismatic cohomology. This is based on recent joint work with Ben Bakker, Abhishek Oswal, and Ananth Shankar.