# Excursions in Arithmetic Geometry

## AMS Annual Meeting Special Session

### Boston, January 4-7, 2023

Organizer

Tony Shaska ([email protected])

### Schedule

Hynes Convention Center on Friday, January 6, from 1:00 PM - 6:00 PM in room 201

Hynes Convention Center on Saturday, January 7, from 8:00 AM - 12:00 PM in room 107

Hynes Convention Center on Saturday, January 7, from 1:00 PM - 6:00 PM in room 107

### Deadline for submitting abstracts

All abstracts must be submitted through the AMS website by **September 13****, **late abstracts *cannot* be accommodated. Please submit abstracts here. Do not forget to pick **Code: SS 9A**

### Overview

Diophantine equations are systems of polynomial equations solved over integers or rational numbers. Diophantine geometry is the study of Diophantine equations using ideas and techniques from algebraic geometry. It is one of the oldest subjects of mathematics and the most popular part of number theory connecting it to algebraic geometry.

The goal of this session is to explore recent developments in the theory of arithmetic geometry and with special focus on curves and Jacobian varieties. We intend to bring together mathematicians, working on this area of research, from the USA and from Europe encouraging further cooperation and discussion. We will especially encourage younger mathematicians and graduate students and newcomers in the area.

The session will focus on the following topics, but we will be open and welcoming to talks which do not fall in the list of topics below.

### Topics of the session include, but are not limited to:

The geometry of curves and Abelian varieties

Endomorphisms and isogenies of Abelian varieties

Galois properties of torsion points and Tate modules

Theory of heights, weighted heights, moduli heights of curves

Height bounds and height conjectures

Zeta functions of algebraic varieties

Hyperelliptic and superelliptic Jacobians, and their endomorphism rings

Brauer group of abelian varieties, K3 surfaces and generalized Kummer varieties

Néron-Tate heights on abelian varieties

Minimal models, models of curves with minimal height

Néron models of Abelian varieties

Brauer-Siegel theorem in arithmetic geometry

Abelian varieties and the Mordell-Lang conjecture

Effective computation of the Mordell-Weil group and set of rational points

Bombieri-Lang conjecture

Vojta’s conjecture

Other topics

### Confirmed speakers

(to be completed)

Hulya Arguz

Dori Bejleri

Ethan Cotterill

Elira Curri

Brendan Hassett

Mee Seong Im

Sajad Salami

Lee Tae Young

Scheduling

We have very limited time from the AMS. In addition, there are several sessions with which this session could overlap in terms of topics and interests. In order to make sure that everybody can attend talks that they want and still give a talk in this session we are asking for your help with the schedule. Please feel the form below:

Abstracts

Hypergeometric Sheaves And Finite General Linear Groups

**Submission Start Date/Time** 2022-07-24 03:17:01

**Subject:** 11T23

**Presenting Author:** Lee Tae Young, [email protected]

In their recent works, Katz, Rojas-León and Tiep have been searching for relatively simple exponential sums which realizes each finite group that can occur as a quotient of the fundamental group of . They classified the finite almost quasisimple groups which can be realized as monodromy groups of hypergeometric sheaves, and for each type of such groups, they constructed an explicit example of hypergeometric sheaf having this group as its monodromy group. The next step of the program is to find all hypergeometric sheaves which realize each of these groups. In this talk, we will briefly review their works, then discuss how to find all hypergeometric sheaves with monodromy groups isomorphic to or its variants using representation theory, and relations to constructions of Abhyankar.

### ID#15130

### Quivers And Curves In Higher Dimension

**Submission Start Date/Time** 2022-07-24 10:28:34

**Subject:** 14N35

**Presenting Author:** Hulya Arguz, [email protected]

Let be a quiver with vertices. The combinatorics of mutations of leads to the construction of a -dimensional Poisson -cluster variety . On the other hand, for any choice of potential , the representation theory of leads to the construction of a 3-dimensional Calabi-Yau category . We prove, under certain assumptions, a correspondence between Donaldson-Thomas invariants of and counts of rational curves given by log Gromov–Witten invariants of normal crossings compactifications of . This is joint work in progress with Pierrick Bousseau.

### ID#15032

### Arithmetic Enumerative Geometry Of Degeneracy Loci For Algebraic Curves

**Submission Start Date/Time** 2022-07-19 23:55:06

**Subject:** 14H51

**Presenting Author:** Ethan Cotterill, [email protected], Changho Han, [email protected] Naizhen Zhang, [email protected]

In classical algebraic geometry, most enumerative problems amount to calculations of cohomology classes of degeneracy loci for maps of vector bundles; and Brill–Noether theory, or the study of linear series on algebraic curves, provides many nice examples of this type. In this talk we will discuss arithmetic enhancements of Brill–Noether-type classes to formulas valued in the Grothendieck–Witt ring of an arbitrary base field, with a focus on inflectionary and secant plane loci.

### ID#15018

### Generators And Splitting Field Of Shioda’s Elliptic Surfaces For

**Submission Start Date/Time** 2022-07-19 19:30:57

**Subject:** 11G05

**Presenting Author:** Sajad Salami, [email protected]

Given an elliptic surface with the generic fiber defined over it is known that the set of -rational points of , denoted by , is a finitely generated abelian group called the Mordell-Weil group of over . The *splitting field* of over is defined to be the smallest subfield of for which . In terms of the Galois theory, is an extension of with the finite Galois group such that the -invariant elements of are the -rational points.

In this talk, I will speak on the results of my current work on determining the splitting field and a set of independent generators for , the Mordell–Weil group of the generic fiber of the Shioda’s famous elliptic surface given by for

### ID#15009

### Height Moduli On Cyclotomic Stacks

**Submission Start Date/Time** 2022-07-19 14:43:37

**Subject:** 14G40

**Presenting Author:** Dori Bejleri, [email protected]

Heights on stacks were recently introduced by Ellenberg, Satriano and Zureick-Brown as a tool to unify and generalize various results and conjectures about counting arithmetic objects. Cyclotomic stacks are a particularly well behaved class of stacks which share many similarities with projective varieties. In this talk I will present a geometric approach to heights on cyclotomic stacks and a construction of moduli spaces of points of fixed height in the function field case. This allows us to lift counts of arithmetic objects to the Grothendieck ring of varieties and use tools from algebraic geometry and topology to count rational points. I will explain some applications to counting elliptic curves over function fields and some general conjectures. This is based on joint work with Park and Satriano.

### ID#14739

### GIT Heights And Weighted Heights

**Submission Start Date/Time** 2022-06-25 09:02:41

**Subject:** 11G50

**Presenting Author:** Elira Curri, [email protected]

We will explore how the GIT height or invariant height is related to the weighted moduli height . Some computational results will be presented. The talk will be directed to a general audience.