NTS ABSTRACTSpring2024

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Jan 25

Jason Kountouridis

The monodromy of simple surface singularities in mixed characteristic

Given a smooth proper surface X over a $p$-adic field, it is a generally open problem in arithmetic geometry to relate the mod $p$ reduction of X with the monodromy action of inertia on the $\ell$-adic cohomology of X, the latter viewed as a Galois representation ($\ell \neq p$). In this talk, I will focus on the case of X degenerating to a surface with simple singularities, also known as rational double points. This class of singularities is linked to simple simply-laced Lie algebras, and in turn this allows for a concrete description of the associated local monodromy. Along the way we will see how Springer theory enters the picture, and we will discuss a mixed-characteristic version of some classical results of Artin, Brieskorn and Slodowy on simultaneous resolutions.

Feb 01

Brian Lawrence

Conditional algorithmic Mordell

Conditionally on the Fontaine-Mazur, Hodge, and Tate conjectures, there is an algorithm that finds all rational points on any curve of genus at least 2 over a number field. The algorithm uses Faltings's original proof of the finiteness of rational points, along with an analytic bound on the degree of an isogeny due to Masser and Wüstholz. (Work in progress with Levent Alpoge.)

Feb 08

Haoyang Guo

Frobenius height of cohomology in mixed characteristic geometry

In complex geometry, it is known that the i-th cohomology of a variation of Hodge structures on a smooth projective complex variety has the weight increased by at most i. In this talk, we consider the mixed and the positive characteristic analogues of this fact. We recall the notion of prismatic cohomology and prismatic crystals introduced by Bhatt and Scholze, and show that Frobenius height of the i-th prismatic cohomology of a prismatic F-crystal behaves the same. This is a joint work with Shizhang Li.

Feb 15

Sachi Hashimoto

p-adic Gross--Zagier and rational points on modular curves

Faltings' theorem states that there are finitely many rational points on a nice projective curve defined over the rationals of genus at least 2. The quadratic Chabauty method makes explicit some cases of Faltings' theorem. Quadratic Chabauty has recent notable success in determining the rational points of some modular curves. In this talk, I will explain how we can leverage information from p-adic Gross--Zagier formulas to give a new quadratic Chabauty method for certain modular curves. Gross--Zagier formulas relate analytic quantities (special values of p-adic L-functions) to invariants of algebraic cycles (the p-adic height and logarithm of Heegner points). By using p-adic Gross--Zagier formulas, this new quadratic Chabauty method makes essential use of modular forms to determine rational points.

Feb 22

Zhiyu Zhang

Asai L-functions and twisted arithmetic fundamental lemmas

Asai L-functions for GLn are related to the arithmetic of Asai motives. The twisted Gan—Gross—Prasad conjecture opens a way of studying (a twist of) central Asai L-values via descents and period integrals. I will consider an arithmetic analog of the conjecture on central derivatives. I will formulate and prove a twisted arithmetic fundamental lemma. The proof is based on new mirabolic special cycles on Rapoport—Zink spaces, and globalization via Kudla—Rapoport cycles and new twisted Hecke cycles on unitary Shimura varieties. This talk is given online. Meeting ID: 930 1493 4562 Passcode: 1814

Feb 29

Andreas Mihatsch

Generating series of complex multiplication points

A classical result of Zagier states that the degrees of Heegner divisors on the modular curve form the positive Fourier coefficients of a modular form. In my talk, I will define complex multiplication cycles on the Siegel modular variety and show that their degrees have a similar modularity property. This is joint work with Lucas Gerth.

March 7

Baying Liu

Recent progress on certain problems related to local Arthur packets of classical groups

In this talk, I will introduce recent progress on certain problems related to local Arthur packets of classical groups. First, I will introduce a joint work with Freydoon Shahidi towards Jiang's conjecture on the wave front sets of representations in local Arthur packets of classical groups, which is a natural generalization of Shahidi's conjecture, confirming the relation between the structure of wave front sets and the local Arthur parameters. Then, I will introduce a joint work with Alexander Hazeltine and Chi-Heng Lo on the intersection problem of local Arthur packets for symplectic and split odd special orthogonal groups, with applications to the Enhanced Shahidi's conjecture, the closure relation conjecture, and the conjectures of Clozel on unramified representations and automorphic representations. This intersection problem also has been worked out independently at the same time by Hiraku Atobe. In a recent joint work with Alexander Hazeltine, Chi-Heng Lo, and Freydoon Shahidi, we made an upper bound conjecture on wavefront sets of admissible representations of connected reductive groups. If time permits, I will also introduce our recent progress towards this conjecture and its connection with Jiang's conjecture.

Mar 14

Peter Humphries

Restricted Arithmetic Quantum Unique Ergodicity

The quantum unique ergodicity conjecture of Rudnick and Sarnak concerns the mass equidistribution in the large eigenvalue limit of Laplacian eigenfunctions on negatively curved manifolds. This conjecture has been resolved by Lindenstrauss when this manifold is the modular surface assuming these eigenfunctions are additionally Hecke eigenfunctions, namely Hecke-Maass cusp forms. I will discuss a variant of this problem in this arithmetic setting concerning the mass equidistribution of Hecke-Maass cusp forms on submanifolds of the modular surface, along with connections to period integrals of automorphic forms.

Mar 21

Wanlin Li

The nontriviality of the Ceresa cycle

The Ceresa cycle is an algebraic 1-cycle in the Jacobian of a smooth algebraic curve with a chosen base point. It is algebraically trivial for a hyperelliptic curve and non-trivial for a very general complex curve of genus at least 3. Given a pointed algebraic curve, there is no general method to determine whether the Ceresa cycle associated to it is rationally or algebraically trivial. In this talk, I will discuss some methods and tools to study this problem.

Apr 04

Frank Calegari

The arithmetic of some Dirichlet L-values

Starting with the "Leibniz" formula for π: π/4 = 1 - 1/3 + 1/5 - 1/7 + ..., the special values of Dirichlet L-functions have long been a source of fascination and frustration. From Euler's solution in 1734 of the Basel problem to Apery's proof in 1978 that $\zeta(3)$ is irrational, our progress on understanding the arithmetic of these numbers has been limited. In this talk, we discuss some results (old and new) about these numbers.

Apr 11

Sean Howe

Bi-analytic structures on moduli spaces in p-adic Hodge theory

Infinite level basic local Shimura varieties are p-adic moduli spaces that parameterize structures arising in the p-adic cohomology of p-adic varieties. They have long been of interest to number theorists because their cohomology realizes a part of the local Langlands correspondence. To define them one must leave the world of rigid analytic spaces, which is the part of p-adic geometry that we expect to behave like the more familiar theory of complex analytic manifolds, and enter the fractal realm of perfectoid spaces and diamonds. Nonetheless, each infinite level basic local Shimura variety can be constructed in two distinct ways as an inverse limit of a tower of finite covering maps between rigid analytic varieties --- a classical example of this phenomenon is the isomorphism between the Lubin-Tate and Drinfeld towers of moduli of formal groups. In this talk, we will discuss the interaction between these two analytic structures. In particular, we will explain a p-adic Ax-Lindemann theorem (joint with Klevdal) that implies the bi-analytic subvarieties are precisely the special subvarieties, i.e. those subvarieties parameterizing object with extra symmetries. We will also state a cohomological finiteness conjecture generalizing work of Ivanov and Weinstein.

Apr 18

Max Wenqiang Xu

Real zeros of Fekete Polynomials and positive definite characters

In 1911, Fekete proposed the problem of studying how likely a Fekete polynomial has no real zeros in [0,1]. The problem has attracted a lot of attention and a series of work has been done. Notably, the work of Baker and Montgomery in 1989 qualitatively showed that Fekete polynomials without real zeros in [0,1] are rare. In a joint work in progress with Angelo and Soundararajan, we give a quantitative upper bound which is close to the conjectural bound.

Apr 25

Ivan Aidun

Arithmetic Strength of Curves

The strength of a homogeneous polynomial f is the least number of reducible forms g_i h_i needed in an expression of the form f = g_1 h_1 + g_2 h_2 + .... The collective strength of several homogeneous polynomials is the minimal strength of any homogeneous linear combination. These invariants have recently received great attention in the commutative algebra world in connection to progress on Stillman's conjecture. Jordan observed that over a non-algebraically closed field, the existence of a rational solution to f = 0 implies a certain kind of strength decomposition, suggesting that it may be profitable to ask "rational points" types of questions about the existence of strength decompositions. In this talk, I will discuss some of what is known about strength, and some partial progress towards understanding strength of polynomials over "arithmetically interesting" fields.

May 02

Jordan Ellenberg

Non-deterministic diophantine equations

In my work with Will Hardt on Smyth’s conjecture, we found that this old question in algebraic number theory could be expressed as a Diophantine problem in probability distributions. This notion of “non-deterministic diophantine equation” leads to a whole family of questions I basically know nothing about and which I think pose an interesting direction for number theorists. An example of such a system of equations is X = Y = X+Y where X,Y are random variables and “=” here means “equal in distribution.” Thought of as a classical equation in P^1, this equation has no solutions; but it does have a solution in a nonzero joint distribution on X,Y. (Can you think of one? Can you think of one which is supported on a finite subset of Z^2?) I’ll say what this has to do with eigenvalues of linear combinations of permutation matrices and describe the non-deterministic local to global principle which is at the heart of our work on Smyth’s conjecture.