NTS ABSTRACTSpring2023
Jan 26
| MSRI/SLMath workshop |
| Introductory Workshop: Algebraic Cycles, L-Values, and Euler Systems |
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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 |
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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.
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Feb 09
| MSRI/SLMath workshop |
| Introductory workshop: Diophantine Geometry |
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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 |
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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 |
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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 |
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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 |
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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. |