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.

March 02

MSRI/SLMath workshop

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.