NTS ABSTRACTSpring2023

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Jan 26

MSRI/SLMath workshop

Introductory Workshop: Algebraic Cycles, L-Values, and Euler Systems

NTS of the week is cancelled as most of the number theory group are attending the MSRI/SLMath Introductory Workshop: Algebraic Cycles, L-Values, and Euler Systems, see https://www.msri.org/workshops/979.

Feb 02

Asvin Gothandaraman

A p-adic Chebotarev density theorem and functional equation

We (Asvin G, Yifan Wei and John Yin) define a notion of splitting density for "nice" generically finite maps over p-adic fields and show that these densities satisfy a functional equation. As a consequence, we prove a conjecture about factorization probabilities of Bhargava, Cremona, Fisher, Gajovic. Zoom ID: 93014934562 Password: The order of A9 (the alternating group of 9 elements)

Feb 09

MSRI/SLMath workshop

Introductory workshop: Diophantine Geometry

NTS of the week is cancelled as most of the number theory group are attending the MSRI/SLMath introductory workshop on Diophantine Geometry, see https://www.msri.org/workshops/977.

Feb 16

Qirui Li

The linear AFL for non-basic locus

The Arithmetic Fundamental Lemma (AFL) is a local conjecture motivated by decomposing both sides of the Gross—Zagier Formula into local terms using the Relative Trace formula. For each of the local terms, one side is the intersection number in some Rappoport—Zink space. The other side is some orbital integral. To reduce the global computation to local, one needs to consider intersection numbers on both basic and non-basic locus, while the original linear AFL only considers basic locus. Collaborated with Andreas Mihatsch, we consider the non-basic locus of Unitary Shimura varieties and conjectured a similar version of linear AFL for Rappoport Zink space on non-basic locus parameterizing p-divisible groups with étale extensions. We proved that this version of linear AFL conjecture can be essentially reduced to the linear AFL conjecture for Lubin—Tate spaces, which corresponds to the basic locus parameterizing one-dimensional connected p-divisible groups.

Feb 23

Tony Feng

Modularity of higher theta functions

Classical theta functions are generating functions for counting vectors in a lattice. They turn out to have a miraculous symmetry property called modularity, which is proved by some simple (by modern standards) Fourier analysis. Kudla discovered analogs of theta functions in arithmetic geometry, called arithmetic theta functions, which are generating functions composed of algebraic cycles in moduli spaces. These are also expected to enjoy modularity, but this is unknown in most cases and has been very difficult in the known cases. In joint work with Zhiwei Yun and Wei Zhang, we construct “higher” theta functions in the function field context going beyond classical and arithmetic theta functions, and we prove a modularity property in total generality.

March 02

Naser Talebizadeh Sardari

Limiting distributions of conjugate algebraic integers

Let $\Sigma \subset \mathbb{C}$ be a compact subset of the complex plane, and $\mu$ be a probability distribution on $\Sigma$. We give necessary and sufficient conditions for $\mu$ to be the weak* limit of a sequence of uniform probability measures on a complete set of conjugate algebraic integers lying eventually in any open set containing $\Sigma$. Given $n\geq 0$, any probability measure $\mu$ satisfying our necessary conditions, and any open set $D$ containing $\Sigma$, we develop and implement a polynomial time algorithm in $n$ that returns an integral monic irreducible polynomial of degree $n$ such that all of its roots are inside $D$ and their root distributions converge weakly to $\mu$ as $n\to \infty$. We also prove our theorem for $\Sigma\subset \mathbb{R}$ and open sets inside $\mathbb{R}$ that recovers Smith's main theorem~\cite{Smith} as special case. Given any finite field $\mathbb{F}_q$ and any integer $n$, our algorithm returns infinitely many abelian varieties over $\mathbb{F}_q$ which are not isogenous to the Jacobian of any curve over $\mathbb{F}_{q^n}$.

March 09

Carsten Peterson

Quantum ergodicity on Bruhat-Tits buildings

Quantum ergodicity concerns equidistribution properties of eigenfunctions of Laplace-like operators on geometric spaces for which some associated geometric flow is ergodic (such as the geodesic flow on a hyperbolic surface). More recently several authors have investigated quantum ergodicity for sequences of spaces which converge (in the sense of Benjamini-Schramm) to their common universal cover (such as a sequence of hyperbolic surfaces whose injectivity radii go to infinity). Previous authors have considered the settings of regular graphs, rank one symmetric spaces, and some higher rank symmetric spaces. We prove analogous results in the case when the underlying common universal cover is the Bruhat-Tits building associated to $PGL(3, F)$ where $F$ is a non-archimedean local field. This may be seen as both a higher rank analogue of the regular graphs setting as well as a non-archimedean analogue of the symmetric space setting.

March 16

MSRI/SLMath workshop

Shimura Varieties and L functions

NTS of the week is cancelled as most of the number theory group are attending the MSRI/SLMath workshop on Shimura Varieties and L functions, see https://www.msri.org/workshops/1032.

March 23

The eigencurve over the boundary of the weight space

Zijian Yao

The eigencurve is a rigid analytic curve that p-adically interpolates eigenforms of finite slope. The global geometry of the eigencurve is somewhat mysterious. However, over the boundary, it is predicted to behave rather nicely (by the so-called Halo conjecture). This conjecture has been studied by Liu--Wan--Xiao for definite quaternion algebras. In this talk, I will discuss this conjecture in the case of GL2. If time permits, we will discuss some generalizations towards groups beyond GL2. This is joint with H. Diao.

March 30

Higher Modularity of Elliptic Curves

Jared Weinstein

Elliptic curves E over the rational numbers are modular: this means there is a nonconstant map from a modular curve to E. When instead the coefficients of E belong to a function field, it still makes sense to talk about the modularity of E (and this is known), but one can also extend the idea further and ask whether E is 'r-modular' for r=2,3.... To define this generalization, the modular curve gets replaced with Drinfeld's concept of a 'shtuka space'. The r-modularity of E is predicted by Tate's conjecture. In joint work with Adam Logan, we give some classes of elliptic curves E which are 2- and 3-modular.

April 6

Slopes of modular forms and geometry of eigencurves

Nha Xuan Truong

The slopes of modular forms are the $p$-adic valuations of the eigenvalues of the Hecke operators $T_p$. The study of slopes plays an important role in understanding the geometry of the eigencurves, introduced by Coleman and Mazur. The study of the slope began in the 1990s when Gouvea and Mazur computed many numerical data and made several interesting conjectures. Later, Buzzard, Calegari, and other people made more precise conjectures by studying the space of overconvergent modular forms. Recently, Bergdall and Pollack introduced the ghost conjecture that unifies the previous conjectures in most cases. The ghost conjecture states that the slope can be predicted by an explicitly defined power series. We prove the ghost conjecture under a certain mild technical condition. This is joint work with Rouchuan Liu, Liang Xiao, and Bin Zhao.

April 13

A Relative Trace Formula on $\mathrm{GL}(n+1)$ and Its Arithmetic Consequences

Liyang Yang

We will present a new relative trace formula on $\mathrm{GL}(n+1)$, which is weighted by cusp forms on $\mathrm{GL}(n)$ over number fields. The formula combines the spectral side, an average of Rankin-Selberg L-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ over the full spectrum, with the geometric side, consisting of Rankin-Selberg L-functions for $\mathrm{GL}(n)\times\mathrm{GL}(n)$ and certain explicit meromorphic functions. The resulting formula yields new insights and results towards the central $L$-values for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ and opens up new possibilities for arithmetic applications. We provide some examples of such applications.

April 20

On the orthogonal Kudla conjecture over totally real fields

Jiacheng Xia

On a modular curve, Gross--Kohnen--Zagier proves that certain generating series of Heegner points are modular forms of weight 3/2 valued in the Jacobian. Such a result has been extended to orthogonal Shimura varieties over totally real fields by Yuan--Zhang--Zhang for special Chow cycles assuming absolute convergence of the generating series. Based on the method of Bruinier--Raum over the rationals, we plan to fill this gap of absolute convergence over totally real fields. In this talk, I will lay out the setting of the problem and explain some of the new challenges that we face over totally real fields. This is a joint work in progress with Qiao He.

April 27

MSRI/SLMath workshop

Degeneracy of Algebraic Points

NTS of the week is cancelled as most of the number theory group are attending the MSRI/SLMath workshop on Degeneracy of Algebraic Points, see https://www.msri.org/workshops/1040.

May 4

Uniformity in Mordell—Lang for higher dimensions

Tangli Ge

In this talk, I will introduce the four main ingredients of the proof of the uniform Mordell--Lang conjecture in higher dimensions. Keeping the ingredients in a black box, I will briefly sketch a philosophical proof by Vojta’s method about how these things fit together, with a stress on some of the key differences in higher dimensions compared to curves. This is joint work with Ziyang Gao and Lars Kühne.