Arithmetic and Representation Theory (ART) Seminar

Abstract: In 2001, Conrey, Farmer, Keating, Rubinstein, and Snaith developed a "recipe" utilizing heuristic arguments to predict the asymptotics of moments of various families of L-functions. This heuristic was later extended by Andrade and Keating to include moments and ratios of the family of L-functions associated to hyperelliptic curves of genus g over a fixed finite field. 

In joint work with Bergström, Petersen, and Westerland, we related the moment conjecture of Andrade and Keating to the problem of understanding the homology of the braid group with symplectic coefficients. We computed the stable homology groups of the braid groups with these coefficients, together with their structure as Galois representations, and showed that the answer matches the number-theoretic predictions. Our results, combined with a recent homological stability theorem of Miller, Patzt, Petersen, and Randal-Williams, imply the conjectured asymptotics for all moments in the function field case, for all large enough odd prime powers q.

Abstract: The relative Langlands program of Ben-Zvi–Sakalleridis-Venkatesh offers a framework for describing the category of sheaves on the loop space LX of a G-variety X, where G is a connected reductive algebraic group. Their conjectures generalize the celebrated geometric Satake equivalence of Ginzburg and Mirković–Vilonen. I will survey their ideas and, building on work of Ginzburg–Riche, describe the case of X = G/L, where L is a Levi subgroup of G. 

Abstract:  According to the relative Langlands functoriality conjecture, an admissible morphism between the L-groups of spherical varieties should induce a functorial transfer of the corresponding local and global automorphic spectra. Via the relative trace formula approach, two basic problems are the fundamental lemma and the local transfer on the geometric side of the relative trace formulas. In this talk, we will consider the rank-one spherical variety case, where the admissible morphism between the L-groups is the identity morphism, in which case, Y. Sakellaridis has already established the local transfer. If time permits, we will also talk about some global aspects.

Abstract: I will report on my joint work in progress with Lue Pan which shows that, in the rational p-adic completed cohomology of a general Shimura variety, "sufficiently regular" infinitesimal weights (whose meaning will be explained) can only show up in the middle degree.  I will also mention some byproducts and explain the main ingredients in our work, if time permits.

Abstract: We construct the integral Sen operator and various (Nygaard, conjugate, Hodge) filtrations attached to integral crystalline representations, which Gee-Kisin and Bhatt-Gee-Kisin use to understand the reduction of crystalline representations. Our methods are more explicit and allow us to extend their constructions to more general settings.  This is a joint work with Hui Gao. 

Abstract: For any smooth projective curve over a finite field, we construct the p-adic analytic moduli stack of isocrystals and study its geometry. This is the crystalline analogue of the moduli of integrable connections. Notably, even though it is a characteristic 0 object, the moduli stack admits a Frobenius pullback endomorphism. We will explain motivations coming from the global Langlands correspondence, and illustrate how the geometry of the moduli can be used to count (the p-adic analogues of) local systems. Joint work with Koji Shimizu.

Abstract: We discuss the impacts of the progress in automorphic forms to the representation theory of reductive algebraic groups over local fields. In particular the impacts of the endoscopic classification of the discrete automorphic spectrum for classical groups by J. Arthur and other followers to the problems on the wavefront sets and the unitary dual. This will be an overview of my recent work, joint with A. Hazeltine, B. Liu, C.-H. Lo and Q. Zhang, with B. Liu, C.-H. Lo and L. Mason-Brown, and with D. Liu and L. Zhang. More accessible introduction and details will be provided in the Math 8280, Spring, 2026. 

Abstract: In this talk, we will motivate and explain Scholze’s Igusa stack conjecture, which predicts the existence of a v-stack (the Igusa stack) which interpolates the various Igusa varieties associated with a given Shimura variety. We will report on progress on the conjecture in the Hodge and abelian-type cases, and we will discuss some of the (many) consequences for the study of the cohomology of Shimura varieties. This talk is based on joint work with Pol van Hoften, Dongryul Kim, and Mingjia Zhang.

Abstract:  I discuss my joint work with Jingwei Xiao and Wei Zhang, where we formulate a global conjecture relating periods associated to certain symmetric spaces of unitary groups to central values of standard L-functions on linear groups, generalizing a theorem of Waldspurger for GL(2). To attack this conjecture, we introduce a new family of relative trace formulas which we successfully compare under certain local assumptions to prove cases of the conjecture. A new feature of this comparison is the presence of relative endoscopy: we need to affect an endoscopic comparison of relative trace formulas to establish the global results. All of this relies on establishing several new local results.