NTS ABSTRACTSpring2025

Back to the number theory seminar main webpage: Main page

Jan 30

L-values and the Mahler measures of polynomials

Xuejun Guo (Nanjing University)

When the zero locus of a tempered polynomial f(x,y) defines an elliptic curve E, the  value L(E,2) is related to the Mahler measure of  f(x,y). In this talk,  we will explore explicit identities that connect these L-values with Mahler measures for several families of polynomials.

Feb 6

The Geometry and Combinatorics of Matrix Points on Curves

Yifan Wei (UW-Madison)

The study of matrix points on curves was initiated by Yifeng Huang in his consideration of Cohen-Lenstra series, which captures the groupoid point-counts of the number of commuting matrices solutions to the defining equation of a curve. In our recent work, we incorporated ideas from geometric representation theory to study the rational cohomology ring of the space of matrix solutions. This new work refines previous results on point-counts, and leads to interesting directions in algebraic combinatorics.

Feb 13

A new proof of the arithmetic Siegel-Weil formula

Joey Yu Luo (UW-Madison)

The arithmetic Siegel-Weil formula establishes a profound connection between intersection numbers in Shimura varieties and the Fourier coefficients of central derivatives of Eisenstein series. This result was proven by C. Li and W. Zhang in 2021 using local methods. In this talk, I will present a new proof of the formula that uses the local-global compatibility and the modularity of generating series of special divisors.

Feb 20

(A)FL at infinity and generating series of arithmetic CM cycles

Tonghai Yang (UW-Madison)

In this talk, we propose a FL and AFL at the real place with a proof of FL (if time permits). We also define a generating series of arithmetic CM cycles indexed by integer and conjecture it to be modular. Finally, we explain the connection between the two. This is a preliminary report of my joint work with Andreas Mihatsch and Siddarth Sankaran.

Mar 13

Criteria for pseudorepresentations to arise from genuine representations

Jinyue Luo (UChicago)

In application to modularity, one often seeks for an R=T theorem, that is, the deformation ring R is isomorphic to a (localized) Hecke algebra T. However, sometimes only the framed deformation ring exists. With the framing variables, it is obviously larger than the Hecke algebra. Pseudorepresentations, which is a generalization of the notion of traces of representations, was invented to get around this issue. We will introduce the notion of pseudorepresentations and discuss the criteria for pseudo representations to arise from genuine representations. Next, we will introduce the algorithm used to explicitly compute usual deformation rings and pseudodeformation rings for finitely presented groups, which leads to the discovery of a counterexample.

Mar 20

Characterization of sub-torii in positive characteristic

Ananth Shankar (Northwestern)

The Manin-Mumford conjecture posits that an irreducible complex subvariety of a torus that contains a Zariski-dense set of torsion points must be the translate of a sub-torus by a torsion point. This conjecture is false in positive characteristic, as every point defined over a finite field is a torsion point. In this talk, I will report on ongoing work with Anup Dixit where we attempt to formulate and prove analogues of this conjecture in positive characteristic.

Apr 3

Nonabelian Fourier kernels on SL_2 and GL_2

Zhilin Luo (UChicago)

I will present joint work with B.C. Ngô unveiled an explicit formula for nonabelian Fourier kernels in G=SL_2 and GL_2 that is conjectured by A. Braverman and D. Kazhdan. I will also explain the connection with the orbital Hankel transform suggested by Ngô.

Apr 10

Kakeya-Nikodym norms of Maass forms

Jiaqi Hou (UW-Madison)

I will talk about the problems on bounding L^p norms of Laplace eigenfunctions on Riemannian manifolds, and, in particular, Maass forms on locally symmetric spaces. The classical result was proved by Sogge, which is sharp on spheres but is expected to be improved with some curvature assumption. When p is small,  Blair and Sogge showed that the problem can be reduced to studying the Kakeya-Nikodym norms, which measure the concentration near geodesic tubes. Using the amplification method, we can prove some power savings for Kakeya-Nikodym norms of Hecke-Maass forms in some cases. In this talk, I will present the results for SL(2,C) and U(2,1).

Apr 17

Random links and arithmetic statistics

Tomer Schlank (UChicago)

Étalé homotopy is a theory that enables the study of algebraic and arithmetic concepts through geometric perspectives. Barry Mazur observed that, from this viewpoint, square-free integers are analogous to links in three-dimensional space. We will explore this analogy and propose a way to give it a statistical flavor. Informally, we assert that a random square-free integer of size approximately X resembles the closure of a random braid on roughly logX strands. We will make this statement more precise by introducing a number-theoretical analog for a family of numerical link invariants (called Kei's) and use this analogy to propose a conjecture regarding their asymptotic behavior. We will also present a few cases where this conjecture has been proven. This is a joint work with Ariel Davis.

Apr 24

Reconstructing genus 4 curves and applications

Sam Schiavone (MIT)

We present a method for recovering the canonical model of a genus 4 curve from its theta constants. We describe some applications, such as gluing genus 2 curves, computing examples of explicit modularity for RM abelian varieties, and computing CM Jacobians. As a final example, we discuss work in progress toward explicitly computing an abelian 4-fold of Mumford type. Joint work with Thomas Bouchet, Jeroen Hanselman, and Andreas Pieper.

May 1

Near-center derivatives and arithmetic 1-cycles

Ryan Chen (MIT)

Theta series for lattices count lattice vectors of fixed norm. Such theta series give some of the first examples of automorphic forms. It is possible to form "theta series" in other geometric contexts, e.g. for counting problems involving abelian varieties. It is expected that these theta series again have additional automorphic symmetry. I will explain some “near-central” instances of an arithmetic Siegel--Weil formula from Kudla’s program. These "geometrize" the classical Siegel--Weil formulas, on lattice and lattice vector counting via Eisenstein series. At these near-central points of functional symmetry, we observe that both the "leading" special value (complex volumes) and the "subleading" first derivative (arithmetic volume) simultaneously have geometric meaning. The key input is a new "limit phenomenon" relating positive characteristic intersection numbers and heights in mixed characteristic, as well as its automorphic counterpart.