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Jan 28

Monica Nevins
Interpreting the local character expansion of p-adic SL(2)

The Harish-Chandra—Howe local character expansion expresses the character of an admissible representation of a p-adic group G as a linear combination with complex coefficients of the (finitely many!) Fourier transforms of nilpotent orbital integrals \(\widehat{\mu}_{\mathcal{O}}\) --- near the identity. Approaching from another direction: we can restrict the representation to any compact open subgroup K of G, obtaining its branching rules, which also describe the representation near the identity, in a different sense. We show that for G=SL(2,k), k a nonarchimedean local field, where the branching rules to maximal compact open subgroups K are known, each of these terms \(\widehat{\mu}_{\mathcal{O}}\) can be interpreted as the character \(\tau_{\mathcal{O}}\) of a representation of K, up to an error term arising from the zero orbit. Moreover, the irreducible components of \(\tau_{\mathcal{O}}\) are explicitly constructed from the K -orbits in \(\mathcal{O}\). This work in progress offers a conjectural alternative interpretation of branching rules of admissible representations.

Feb 4

Ke Chen
On CM points away from the Torelli locus

Coleman conjectured in 1980's that when g is an integer sufficiently large, the open Torelli locus T_g in the Siegel modular variety A_g should contain at most finitely many CM points, namely Jacobians of general curves of high genus should not admit complex multiplication. We show that certain CM points do not lie in T_g if they parametrize abelian varieties isogeneous to products of simple CM abelian varieties of low dimension. The proof relies on known results on Faltings height and Sato-Tate equidistributions. This is a joint work with Kang Zuo and Xin Lv.

Feb 11

Dmitry Gourevitch
Relations between Fourier coefficients of automorphic forms, with applications to vanishing and to Eulerianity

In recent works with H. P. A. Gustafsson, A. Kleinschmidt, D. Persson, and S. Sahi, we found a way to express any automorphic form through its Fourier coefficients, using adelic integrals, period integrals and discrete summation – generalizing the Piatetski-Shapiro – Shalika decomposition for GL(n). I will explain the general idea behind our formulas, and illustrate it on examples. I will also show applications to vanishing and Eulerianity of Fourier coefficients.

Feb 18

Eyal Kaplan
The generalized doubling method, multiplicity one and the application to global functoriality

One of the fundamental difficulties in the Langlands program is to handle the non-generic case. The doubling method, developed by Piatetski-Shapiro and Rallis in the 80s, pioneered the study of L-functions for cuspidal non-generic automorphic representations of classical groups. Recently, this method has been generalized in several aspects with interesting applications. In this talk I will survey the different components of the generalized doubling method, describe the fundamental multiplicity one result obtained recently in a joint work with Aizenbud and Gourevitch, and outline the application to global functoriality. Parts of the talk are also based on a collaboration with Cai, Friedberg and Ginzburg.

Feb 25

Roger Van Peski
Random matrices, random groups, singular values, and symmetric functions

Since the 1989 work of Friedman-Washington, the cokernels of random p-adic matrices drawn from various distributions have provided models for random finite abelian p-groups arising in number theory and combinatorics, the most famous being the class groups of quadratic imaginary number fields. Since any finite abelian p-group is isomorphic to a direct sum of cyclic groups $\bigoplus_i \mathbb{Z}/p^{\lambda_i}\mathbb{Z}$, it is equivalent to study the random integer partition $\lambda = (\lambda_1, \lambda_2,\ldots)$, which is analogous to the singular values of a complex random matrix. We show that the behavior of such partitions under taking products and corners of random p-adic matrices is governed by the Hall-Littlewood polynomials, recovering and explaining some previous results relating p-adic matrix cokernels to these polynomials. We use these exact results to study the joint asymptotic behavior of the cokernels of products of many random p-adic matrices $A_\tau \cdots A_1$, with $\tau$ acting as a discrete time parameter. We show that the parts $\lambda_i$ of the corresponding partition have a simple description via an interacting particle system, and their fluctuations converge under rescaling to independent Brownian motions. At both the exact and asymptotic level we explain connections between our results and existing results on singular values of complex random matrices: both are in fact degenerations of the same operations on random partitions coming from Macdonald polynomials.

Mar 4

Amos Nevo
Intrinsic Diophantine approximation on homogeneous algebraic varieties

Classical Diophantine approximation quantifies the denseness of the set of rational vectors in their ambient Euclidean space. A far-reaching extension of the classical theory calls for quantifying the denseness of rational points in general homogeneous algebraic varieties. This was raised as an open problem by Serge Lang already half a century ago, but progress towards it was achieved only in a limited number of special cases. A systematic approach to this problem for homogeneous varieties associated with semi-simple groups has been developed in recent years, in joint work with A. Ghosh and A. Gorodnik. The methods employ dynamical arguments and effective ergodic theory, and employ spectral estimates in the automorphic representation of semi-simple groups. In the case of homogeneous spaces with semi-simple stability group, this approach leads to the derivation of pointwise uniform and almost sure Diophantine exponents, as well as analogs of Khinchin's and W. Schmidt's theorems, with some of the results being best possible. We will explain some of the main results and some of the ingredients in their proof, focusing on some easily accessible examples.

Mar 11

Carlo Pagano
On the negative Pell conjecture
The negative Pell equation has been studied since many centuries. Euler already provided an interesting criterion in terms of continued fractions. In 1995 Peter Stevenhagen proposed a conjecture for the frequency of the solvability of this equation, when one varies the real quadratic field. I will discuss an upcoming joint work with Peter Koymans where we establish Stevenhagen's conjecture.

Mar 18

Siddhi Pathak
Special values of L-series with periodic coefficients
A crucial ingredient in Dirichlet's proof of infinitude of primes in arithmetic progressions is the non-vanishing of $L(1,\chi)$, for any non-principal Dirichlet character $\chi$. Inspired by this result, one can ask if the same remains true when $\chi$ is replaced by a general periodic arithmetic function. This problem has received significant attention in the literature, beginning with the work of S. Chowla and that of Baker, Birch and Wirsing. Nonetheless, tantalizing questions such as the conjecture of Erdos regarding non-vanishing of $L(1,f)$, for certain periodic $f$, remain open. In this talk, I will present various facets of this problem and discuss recent progress in generalizing the theorems of Baker-Birch-Wirsing and Okada.

Mar 25

Emmanuel Kowalski
Remembrances of polynomial values: Fourier's way
The talk will begin by a survey of questions about the value sets of

polynomials over finite fields. We will then focus in particular on a new phase-retrieval problem for the exponential sums associated to two polynomials; under suitable genericity assumptions, we determine all solutions to this problem. We will attempt to highlight the remarkably varied combination of tools and results of algebraic geometry, group theory and number theory that appear in this study.

(Joint work with K. Soundararajan)

Apr 1

Abhishek Oswal
A non-archimedean definable Chow theorem
In recent years, o-minimality has found some striking applications to diophantine geometry. The utility of o-minimal structures originates from the remarkably tame topological properties satisfied by sets definable in such structures. Despite the rigidity that it imposes, the theory is sufficiently flexible to allow for a range of analytic constructions. An illustration of this `tame' property is the following surprising generalization of Chow's theorem proved by Peterzil and Starchenko - A closed analytic subset of a complex algebraic variety that is also definable in an o-minimal structure, is in fact algebraic. While the o-minimal machinery aims to capture the archimedean order topology of the real line, it is natural to wonder if such a machinery can be set up over non-archimedean fields. In this talk, we shall explore a non-archimedean analogue of an o-minimal structure and a version of the definable Chow theorem in this context.

Apr 8

Henri Darmon
Hilbert’s twelfth problem and deformations of modular forms
Hilbert’s twelfth problem asks for the construction of abelian extensions of number fields via special values of (complex) analytic functions. An early prototype for a solution is the theory of complex multiplication, culminating in the landmark treatise of Shimura and Taniyama which provides a satisfying answer for CM ground fields.

For more general number fields, Stark’s conjecture leads to a conjectural framework for explicit class field theory based on the leading terms of abelian complex L-series at s=1 or s=0. While there has been very little no progress on the original conjecture, Benedict Gross formulated seminal p-adic and ``tame’’ analogues in the mid 1980’s which have turned out to be far more amenable to available techniques, growing out of the proof of the ``main conjectures” by Mazur and Wiles, and culminating in the recent work of Samit Dasgupta and Mahesh Kakde which, by proving a strong refinement of the p-adic Gross-Stark conjecture, leads to what might be touted as a {\em p-adic solution} to Hilbert’s twelfth problem for all totally real fields.

I will compare and contrast this work with a different approach (in collaboration with Alice Pozzi and Jan Vonk) which proves a similar result for real quadratic fields. While limited for now to real quadratic fields, this approach is part of the broader program of extending the full panoply of the theory of complex multiplication to real quadratic, and possibly other non CM base fields.

Apr 15

Joshua Lam
CM liftings on Shimura varieties
I will discuss results on CM liftings of mod p points of Shimura varieties, for example the finiteness of supersingular points admitting CM lifts. I’ll also discuss the structure of CM liftable points in the case of Hilbert modular varieties, by applying results of Helm-Tian-Xiao on the Goren-Oort stratification. This is joint work with Mark Kisin, Ananth Shankar and Padma Srinivasan.

Apr 22

Brian Lawrence
The Shafarevich conjecture for hypersurfaces in abelian varieties
Let K be a number field, S a finite set of primes of O_K, and g a positive integer. Shafarevich conjectured, and Faltings proved, that there are only finitely many curves of genus g, defined over K and having good reduction outside S. Analogous results have been proven for other families, replacing "curves of genus g" with "K3 surfaces", "del Pezzo surfaces" etc.; these results are also called Shafarevich conjectures. There are good reasons to expect the Shafarevich conjecture to hold for many families of varieties: the moduli space should have only finitely many integral points. Will Sawin and I prove this for hypersurfaces in a fixed abelian variety of dimension not equal to 3.

Apr 29

Maria Fox
Supersingular Loci of Some Unitary Shimura Varieties
Unitary Shimura varieties are moduli spaces of abelian varieties with an action of a quadratic imaginary field, and extra structure. In this talk, we'll discuss specific examples of unitary Shimura varieties whose supersingular loci can be concretely described in terms of Deligne-Lusztig varieties. By Rapoport-Zink uniformization, much of the structure of these supersingular loci can be understood by studying an associated moduli space of p-divisible groups (a Rapoport-Zink space). We'll discuss the geometric structure of these associated Rapoport-Zink spaces as well as some techniques for studying them.

May 6

Padmavathi Srinivasan
Towards a unified theory of canonical heights on abelian varieties
p-adic heights have been a rich source of explicit functions vanishing on rational points on a curve. In this talk, we will outline a new construction of canonical p-adic heights on abelian varieties from p-adic adelic metrics, using p-adic Arakelov theory developed by Besser. This construction closely mirrors Zhang's construction of canonical real valued heights from real-valued adelic metrics. We will use this new construction to give direct explanations (avoiding p-adic Hodge theory) of the key properties of height pairings needed for the quadratic Chabauty method for rational points. This is joint work in progress with Amnon Besser and Steffen Mueller.