Excursions in Arithmetic Geometry

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

GIT Heights And Weighted Heights

• Presenting Author: Elira Curri, [email protected]

• Author: Tony Shaska, [email protected]

Historically arithmetic invariant theory was born as part of arithmetic intersection theory and Arakelov theory with attempts of Cornalba/Harris, Burnol, Zhang, et al. GIT height or invariant height was introduced by Zhang for Chow semistable algebraic cycles . We will explore how this height is related to the weighted moduli height in the case of binary forms.

Height Moduli On Cyclotomic Stacks

• 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.

Generators And Splitting Field Of Shioda’s Elliptic Surfaces  For 

• 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 

Arithmetic Enumerative Geometry Of Degeneracy Loci For Algebraic Curves

• Presenting Author: Ethan Cotterill, [email protected]

• Author: Changho Han, [email protected]

• Author: 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.

Hypergeometric Sheaves And Finite General Linear Groups

• 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.

Quivers And Curves In Higher Dimension

• 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.

Isogenous Elliptic Subcovers Of Genus Two Curves.

• Presenting Author: Lubjana Beshaj, [email protected]

Genus 2 curves with (n, n)-decomposable Jacobians are the most studied type of genus 2. Such curves have received new attention lately due to interest on their use on cryptographic applications and their suggested use on post-quantum crypto-systems and random self-reducibility of discrete logarithm problem.

In this talk we show that for N = 2, 3, 5, and 7 there are only finitely many genus two curves C (up to isomorphism) defined over Q with (2, 2)-split Jacobian and automorphism group of order 4, such that their elliptic subcovers are N-isogenous. Also, there are only finitely many genus two curves (up to isomorphism) defined over Q with (3, 3)-split Jacobian such that their elliptic subcovers are 5-isogenous.

Unlikely And Just Likely Intersections On Shimura Varieties

• Presenting Author: Asvin G, [email protected]

There has been much recent interest in the question of "just-likely" intersections on Shimura varieties - i.e.,intersections of complimentary dimension subvarieities up to Hecke translations. Almost all the work so far has concerned the intersection of special divisors with curves. I will present the first results dealing with the intersection of arbitrary subvarities in two cases - Hilbert surfaces over a finite field and a product of modular curves over a wide range of base rings.

In the opposite direction, I will also exhibit surprising examples of "unlikely" intersections contradicting a reasonable heuristic about isogeny classes of abelian varieties over a finite field.

Torsion Of Jacobians For Unramified  Covers Of Curves

• Presenting Author: Douglas L Ulmer, [email protected]

Let  be a perfect field of characteristic , let  be a smooth projective curve over , and let  be an unramified Galois cover with group . We study connections between the de Rham cohomology of  and that of  (with their structures as module over the Dieudonne ring  and their action of ). The cornerstone is a result of Chevalley-Weil type stating that  is close to being a free module over . The motivation for studying this question comes from Oda’s theorem which says that  (the -torsion subgroup scheme of the Jacobian of ) has Dieudonne module . Our results show that  is very close to being a -fold repeated extension of copies of . We also explore the restrictions on  that follow. This is joint work with Bryden Cais.

Isogenous Elliptic Curves In A Family Ordered By Heights

• Presenting Author: Yu Fu, [email protected]

Let  be a rational curve over  isomorphic to  parametrizes a one-dimensional family of pairs of elliptic curves and let  denote the generic fiber of this family over . We give an upper bound of the number of specializations of  such that the two factors are isogenous, with the assumption that the height of the parameter . We exploit results from dimension growth conjectures to get a uniform upper bound that depends only on the degree of  and height .

Subspace Configurations And Low Degree Points On Curves

• Presenting Author: Borys Kadets, [email protected]

A curve  over a number field  is said to have arithmetic irrationality , , if  has infinitely many points of degree  but only finitely many points of degrees less than . For example, a plane conic has arithmetic irrationality  if it admits a rational point, and  otherwise. A simple classification is known for : in these cases  is geometrically a degree  covering of  or an elliptic curve; for  this is due to Faltings’theorem, the case  was handled by Harris and Silverman, and the case  was done by Abramovich and Harris. Surpisingly, the picture changes for : for all  Debarre and Fahlaoui constructed examples of curves  with  and which do not admit degree  maps to other curves.

I will describe some general classification results for curves with arithmetic irrationality . I will explain how the classification problem can be reduced to a study of low genus curves, and as an application describe the full classification for . This relies on analyzing a new discrete-geometric object – the subspace configuration – attached to curves with . This talk is based on joint work with Isabel Vogt (arXiv:2208.01067).

Heights On Stacks

• Presenting Author: Jordan S Ellenberg, [email protected]

The study of rational points on algebraic stacks is by now understood to be part of Diophantine geometry; but one thing that has been missing is a theory of heights for such points. I’ll talk about a definition (joint with Satriano and Zureick-Brown), explain what it has to do with other notions of height, talk about recent work of Darda and Yasuda which generalizes this notion and allows for geometrically motivated conjectures of strong Malle and Batyrev-Manin type, and finally talk about recent work of Nasserden-Xiao, Boggess-Sankar, Yin, and others which provide evidence for these counting conjectures.

Weil Conjectures, Enriched Zeta Functions, And The Topology Of Real Points

• Author: Margaret Bilu, [email protected]

• Author: Wei Ho, [email protected]

• Presenting Author: Padmavathi Srinivasan, [email protected]

• Author: Isabel Vogt, [email protected]

• Author: Kirsten Wickelgren, [email protected]

For a variety defined over the rationals, the Weil conjectures relate the Betti numbers of the set of complex solutions to the number of solutions modulo prime powers, packaged into the zeta function. Can the zeta function also tell us about the Betti numbers of the set of real solutions? In this talk, we will present a quadratic enrichment of the zeta function and show that it naturally encodes the Betti numbers of the real points in some situations. We will also present a rationality result for the enriched zeta function for cellular varieties. This is joint work with Margaret Bilu, Wei Ho, Isabel Vogt, and Kirsten Wickelgren.

Explicit Two-Cover Descent For Genus 2 Curves

• Presenting Author: Daniel Rayor Hast, [email protected]

The Chabauty–Coleman method is an effective -adic approach to computing the set of rational points on genus 2 curves whose Jacobian has Mordell–Weil rank 0 or 1. For higher-rank curves, no fully general method is known. In some such cases, two-cover descent can successfully transfer the problem to a more tractable set of curves. We present an implementation of two-cover descent that, in combination with the elliptic Chabauty method, was applied to a large dataset of genus 2 quintic curves of rank 2 or 3. We demonstrate this method and analyze how often it succeeds in computing the set of rational points and what obstacles lead it to fail in the remaining cases.

Nonsplitting Of The Hilbert Exact Sequence And The Principal Chebotarev Density Theorem

• Presenting Author: Kelly Emmrich, [email protected]

• Author: Lian Duan, [email protected]

• Author: Ning Ma, [email protected]

• Author: Xiyuan Wang, [email protected]

Let  be a finite Galois extension of number fields, and let  be the Hilbert class field of . We find a way to verify the nonsplitting of the short exact sequence

by finite calculation. Our method is based on the study of the principal version of the Chebotarev density theorem, which represents the density of the prime ideals of  that factor into the product of principal prime ideals in . We also find explicit equations to express the principal density in terms of the invariants of . In particular, we prove that the group structure of the ideal class group of  can be determined by reading the principal densities.

Bounds On Torsion Subgroups From Geometric Isogeny Classes Of Elliptic Curves

• Presenting Author: Tyler Genao, [email protected]

A celebrated result of Merel (1996) proved that for each integer  there exists a uniform bound on all torsion subgroups of elliptic curves over number fields of degree ; a priori, such elliptic curves have -invariants of bounded degree over . On the other hand, any -isogeny class of elliptic curves will contain -invariants of arbitrarily large degree. It may be surprising, then, that torsion subgroups in these isogeny classes can still be “well-behaved" while allowing their number field degrees to vary.

In this talk, we will discuss our new results on polynomial bounds for geometric isogeny classes of elliptic curves, as well as special asymptotic bounds (specifically, “typical boundedness") on the collection of geometric isogeny classes with at least one rational -invariant.

K3 Surfaces With Group Actions And Derived Equivalence

• Presenting Author: Brendan E Hassett, [email protected]

• Author: Yuri Tschinkel, [email protected]

Let G be a finite group. We study derived equivalence for K3 surfaces with G-action, with a view toward Diophantine questions over  and other fields. We give applications to how the existence of rational points over isotrivial families behaves under derived equivalence.

On The Periods Of Seiberg-Witten Curves And Connections To Some Abelian Surfaces

• Presenting Author: Michael T. Schultz, [email protected]

In this talk, we will explore some connections between the periods of certain rational elliptic surfaces called Seiberg-Witten curves that arise in physics and the moduli of some abelian varieties. On the Seiberg-Witten side, the (generalized) Picard-Fuchs operators annihilating the periods on the fibres of the rational elliptic surface are realized as a resonant GKZ hypergeometric system that is closely connected to quantities of physical interest, like the RG flow of the underlying supersymmetric quantum gauge theory. On the abelian side, the varieties that appear include hyperelliptic Jacobians, non-PPAS, Pryms, and Kuga-Satake varieties. The connection is based on the familiar quadratic twist construction in the theory of elliptic surfaces, which in this case yields a non-trivial relationship between the Seiberg-Witten curves and the geometry of high Picard rank K3 surfaces. These K3s are tightly connected to Kummer surfaces, and their underlying abelian surfaces via a Shioda-Inose like correspondence. There is then an immediate algebraic relationship between the moduli of these abelian varieties and the physical data of the Seiberg-Witten curve, which we explicitly compute in certain cases. Then, we show how this data can be encoded in terms of the differential geometry on the corresponding moduli space.

Friday Afternoon

1:00-1:45 J. Ellenberg  14G40-18234

2:00-2:20 K. Emmrich 11R45-19383

2:30-2:50 B. Hassett 14J28-20399

3:00-3:45 D. Ulmer 11G20-18010

4:00-4:20 M. Schultz 14K10-21087

4:30-4:50 E.Cotterill 14H51-15032

5:00-5:20 G. Asvin 14G35-17896

5:30-5:50 T. Genao 11G05-19439

Saturday Morning 

9:00-9:20  D. Bejleri  14G40-15009

9:30-9:50 L. T. Young 11T23-15127

10:00-10:20 B. Kadets 11G30-18119

10:30-10:50 H. Arguz 14N35-15130

11:00-11:20 D. Hast 11G30-19079

11:30-11:50 S. Salami 11G05-15018

Saturday afternoon

1:00-1:20  E. Curri 11G50-14739

1:30-1:50 Y. Fu 11G05-18018

2:00-2:20 L. Beshaj 14H99-17823

2:30-2:50 P. Srinivasan 11G25-18808

• Arguz, Hulya

• Asvin, G.

• Bejleri, Dori

• Beshaj, Lubjana

• Cotterill, Ethan

• Curri, Elira

• Ellenberg, Jordan

• Emmrich, Kelly

• Fu, Yu

• Genao, Tyler

• Hassett, Brendan

• Hast, Daniel

• Kadets, Borys

• Salami, Sajad

• Schultz, Michael

• Srinivasan, Padmavathi

• Ulmer, Doug

• Young, Lee Tae