seminars:arit

## The Arithmetic Seminar

TOPICS: Arithmetic in the broadest sense that includes Number Theory (Elementary Arithmetic, Algebraic, Analytic, Combinatorial, etc.), Algebraic Geometry, Representation Theory, Lie Groups and Lie Algebras, Diophantine Geometry, Geometry of Numbers, Tropical Geometry, Arithmetic Dynamics, etc.

PLACE and TIME: This semester the seminar meets on Mondays at 3:30 p.m. or on Tuesdays at 4:15 p.m. in WH 100E, with possible special lectures at other days. Before the talks, there will be refreshments in WH-102.

ORGANIZERS:
Regular Faculy: Alexander Borisov, Marcin Mazur, Adrian Vasiu,
Post-Docs: Vaidehee Thatte.

Current Ph.D. students: Patrick Carney, Andrew Lamoureux, Micah Loverro, Sayak Sengupta, and Changwei Zhou.

Graduated Ph.D. students (in number theory and related topics): Ilir Snopce (Dec. 2009), Xiao Xiao (May 2011), Jinghao Li (May 2015), Ding Ding (Dec. 2015), Patrick Milano (May 2018).

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#### Spring 2019

• January 29 (Tuesday)
Speaker: N/A
Title: Organizational Meeting
Abstract: We will discuss schedule and speakers for this semester
• February 4 (Monday)
Speaker: Xiao Xiao (Utica College)
Title: Automorphism group schemes at finite level of $F$-cyclic $F$-crystals
Abstract: Let $M$ be an $F$-crystal over an algebraically closed field of positive characteristic. For every integer $m \geq 1$, let $\gamma_{M}(m)$ be the dimension of the automorphism group scheme $\mathrm{Aut}_m(M)$ of $M$ at finite level $m$. In 2012, Gabber and Vasiu proved that $0 \leq \gamma_{M}(1) < \gamma_{M}(2) < \cdots < \gamma_{M}(n_{M}) = \gamma_{M}(n_{M}+1) = \cdots$ where $n_{M}$ is the isomorphism number of $M$, and that $\gamma_{M}(m+1)- \gamma_{M}(m) \leq \gamma_{M}(m)- \gamma_{M}(m-1)$ for all $m \geq 1$ if $M$ is a Dieudonn\'e module over $k$. We generalize the same result to arbitrary $F$-crystals in 2014. Questions have been asked whether $\gamma_{M}(m+1)- \gamma_{M}(m) < \gamma_{M}(m)- \gamma_{M}(m-1)$ for all $1 \leq m \leq n_{M}$ for any $F$-crystal $M$. In this talk, we will discuss a combinatorial formula that calculates $\gamma_{M}(m)$ for a certain family of $F$-crystals called $F$-cyclic $F$-crystals. This formula allows to give a negative answer to the aforementioned question in general but a positive answer to some family of Dieudonn\'e modules.
• February 12 (Tuesday)
Speaker: The Talk is canceled
Title: TBA
Abstract: TBA
• February 18/19
Speaker: Alexander Borisov (Binghamton)
Title: An update on the Keller map search
Abstract: TBA
• February 25 (Monday)
Speaker: Liang Xiao (UConn Storrs)
Title: Bloch–Kato conjecture for some Rankin-Selberg motives.
Abstract: The Birch and Swinnerton-Dyer conjecture is known in the case of rank 0 and 1 thanks to the foundational work of Kolyvagin and Gross-Zagier. In this talk, I will report on a joint work in progress with Yifeng Liu, Yichao Tian, Wei Zhang, and Xinwen Zhu. We study the analogue and generalizations of Kolyvagin's result to the unitary Gan-Gross-Prasad paradigm. More precisely, our ultimate goal is to show that, under some technical conditions, if the central value of the Rankin-Selberg L-function of an automorphic representation of U(n)*U(n+1) is nonzero, then the associated Selmer group is trivial; Analogously, if the Selmer class of certain cycle for the U(n)*U(n+1)-Shimura variety is nontrivial, then the dimension of the corresponding Selmer group is one.
• March 4/5
Speaker: Serin Hong (Michigan)
Title: Surjective bundle maps and bundle extensions over the Fargues-Fontaine curve
Abstract: Vector bundles on the Fargues-Fontaine curve play a pivotal role in recent development of p-adic Hodge theory and related fields, as they provide geometric interpretations of many constructions in these fields. The most striking example is the geometrization of the local Langlands correspondence due to Fargues where the correspondence is stated in terms of certain sheaves on the stack of vector bundles on the Fargues-Fontaine curve.
In this talk, we give two classification theorems regarding vector bundles on the Fargues-Fontaine curve: a classification of all pairs of vector bundles with a surjective bundle map between them and a classification of extensions of two given vector bundles satisfying certain conditions. We also explain several applications of our classification theorems, some of which are closely related to the geometrization of the local Langlands correspondence. This talk is based on my recent work plus a previous joint work with C. Birkbeck, T. Feng, D. Hansen, Q. Li, A. Wang and L. Ye.
• March 11/12
Speaker: TBA
Title: TBA
Abstract: TBA
• March 25/26
Speaker: TBA
Title: TBA
Abstract: TBA
• April 1/2
Speaker: TBA
Title: TBA
Abstract: TBA
• April 9 (Tuesday)
Speaker: Renee Bell (UPenn)
Title: TBA
Abstract: TBA
• April 15/16
Speaker: TBA
Title: TBA
Abstract: TBA
• April 22/23
Speaker: TBA
Title: TBA
Abstract: TBA
• April 30 (Tuesday)
Speaker: Evangelia Gazaki (Michigan)
Title: TBA
Abstract: TBA
• May 6/7
Speaker: TBA
Title: TBA
Abstract: TBA