NTS ABSTRACTSpring2023

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.

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 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.

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.