## Group Meetings

Consortium meetings are held roughly every month, and rotate between Leiden, Utrecht and Groningen. They sometimes contain public talks. If you are interested in learning about or attending the public talks in any of the locations, please contact a member of the team for more details.

*Upcoming meetings*

TBA.

*Past meetings*

### May 30th 2024, Leiden

I. Arakelov Theory Seminar: **Tianci Kang **(Groningen)

II. Speaker: **Wim Nijgh** (Leiden)

Title: *Density of rational points on a family of del Pezzo surfaces of degree 1*

Abstract: In this talk, we will discuss the Zariski density of the rational points on a del Pezzo surface X over a field k of degree 1 contained in a certain family. The surfaces that we will look at can be described on an affine open with coordinates 𝑥,𝑦,𝑡 by an equation of the form 𝑦^2=𝑥^3+𝑎(𝑓)𝑥+𝑏(𝑓) where 𝑎,𝑏∈𝑘[𝑢] are polynomials with deg(𝑎)≤1 and deg(𝑏)≤2 and 𝑓∈𝑘[𝑡] is a polynomial of degree 3. We prove that if the characteristic of 𝑘 is zero, then, assuming some relatively mild condition holds for such a surface, the 𝑘-rational points on the surface lie Zariski dense. Conversely, we will also show that in the case that 𝑘 is finitely generated over 𝑄, this condition needs to be fulfilled for the 𝑘-points to lie Zariski dense.

This talk is based on the work of my master thesis under the supervision of Ronald van Luijk.

III. Speaker: **Amie Bray** (Colorado State)

Title:* Counting isogeny classes of Drinfeld modules over finite fields via Frobenius distributions*

Abstract: In 2008, Gekeler expressed the size of an isogeny class of a rank two Drinfeld module over a prime field as a product over primes of local density functions. These local density functions are what one might expect given a random matrix heuristic. In his proof, Gekeler shows that the product of these factors gives the size of an isogeny class by appealing to class numbers of imaginary quadratic orders. Meanwhile, Achter, Altug, Garcia, and Gordon produced a similar product formula for abelian varieties over prime power fields without the calculation of class numbers. Their proof appeals to Kottwitz’s formula for the size of an isogeny class in terms of adelic orbital integrals. Due to Laumon, one can also express the size of an isogeny class of Drinfeld modules over finite fields via adelic orbital integrals. In this talk, we generalize Gekeler’s product formula to higher rank Drinfeld modules over prime power fields by direct comparison to these adelic orbital integrals.

### April 18th 2024, Groningen

I. Arakelov Theory Seminar: **Justin Uhlemann** (Utrecht)

II. Speaker: **Steffen Müller **(Groningen)

Title: *Survey talk on rational points on curves*

Abstract: I will sketch various methods to find the rational points on a curve over the rationals and how to apply them in practice.

III. Speaker:** Ishai Dan-Cohen** (Ben-Gurion)

Title:* Towards Mordellic obstruction devices coming from rational motivic homotopy theory*

Abstract: Let Z be an open subscheme of Spec Z. A smooth scheme X over Z gives rise to a commutative algebra object C*X in the derived category of motives over Z. Some intuition for this rather abstract object comes from the fact that it remembers, for instance, the smooth de Rham complex of the complex analytic space associated to X, regarded as a cdga. On the other hand, if one forgets the algebra structure, then C^*X *lives in a category whose RHom's are governed by algebraic K-theory. One can define a natural filtration of CX by algebras C^i similar in spirit to a Postnikov tower. Following work of I. Iwanari, in simple examples, it's possible to make C^i for small i quite explicit. We can then show that the space of augmentations of C^i has the structure of a finite type affine QQ-scheme. I'll explain how this leads to new proofs of very special cases of Siegel's theorem for curves, and I'll indicate how these techniques may be relevant for studying integral (or rational) points of higher dimensional varieties.

IV. Speaker: **Jaap Top **(Groningen)

Title:* Prime twists of a genus 1 and a genus 2 curve***
**Abstract: Results of A. Genocchi (1855), L. Bastien (1915), P. Monsky (1990), and D.M. Gordon & D. Grant (1993), motivated two master's projects in Groningen. The first, completed in 1998, was by Gert-Jan van der Heiden, now professor of Philosophy in Nijmegen; the second was done in 2018-2019 by Tim Evink who is currently completing his PhD thesis with Jeroen Sijsling in Ulm. The projects provide small steps towards answering the following question: which primes can we written in the form (x-2)(x-1)x(x+1)(x+2) / y^2, for rational numbers x,y?

The analogous question for (x-1)x(x+1)/y^2 asks for the so-called prime congruent numbers. In the talk I intend to give an overview of the results obtained so far.

### March 19th 2024, Utrecht

I. Arakelov theory seminar:** Boaz Moerman** (Utrecht)

II. Speaker: **Lara Vicino** (Groningen)

Title: *Algebraic Geometry codes and Weierstrass semigroups*

Abstract: Algebraic Geometry codes (AG codes) are a family of error-correcting codes introduced by Goppa in the '80s and constructed using algebraic curves defined over a finite field F_q. AG codes, which can be seen as a generalization of Reed-Solomon codes, provide many excellent examples of error-correcting codes. Furthermore, a celebrated result by Tsfasman, Vladut and Zink (1982) showed that, if q is the square of a prime and is larger than or equal to 49, then there exist sequences of AG codes beating the Gilbert-Varshamov bound, for infinitely many values of q. This result is particularly striking, as it means that there are some families of AG codes that are better than random codes.

A central role in the study of the parameters of AG codes is played by the properties of the underlying curve, and especially by Weierstrass semigroups at the points of the curve. Given a point P on an algebraic curve X, the Weierstrass semigroup H(P) is defined as the set of natural numbers n for which there exists a function f on X having pole divisor (f)_\infty = nP. The structure of H(P) in general varies as the point varies, however, it is known that generically the semigroup is the same, and there can exist only finitely many points of X, called Weierstrass points, with a different semigroup. The reasons for interest in Weierstrass semigroups are multifold: on one hand, they hold an intrinsic theoretical interest which arises from Stöhr-Voloch theory, where they are used to obtain characterizing properties of the curve. On the other hand, together with their generalizations to the case of multi-point Weierstrass semigroups, they represent a key ingredient for computing excellent bounds for the minimum distance of AG codes from the curve.

In this talk, I will give an introduction to AG codes, their properties and their connections with Weierstrass semigroups. With this regard, I will present some results concerning the determination of one-point and two-point Weierstrass semigroups at the points of certain maximal curves. In the two-point case, our results lead to the study of new families of two-point AG codes with good parameters from two well-known maximal curves. In the one-point case, our results deal instead with the computation of the Weierstrass semigroup at every point of a maximal curve with the third largest genus. One surprising result is that, unlike what happens for all the other maximal curves where the Weierstrass points are known, the set of Weierstrass points of this curve is much richer than the set of its F_{q^2}-rational points.

Joint work with Peter Beelen, Leonardo Landi, Maria Montanucci and Marco Timpanella.

III. Speaker: **Tian Wang** (Bonn)

Title: *On the distribution of supersingular primes for abelian surfaces*

Abstract:In 1976, Lang and Trotter made a conjecture that predicts the number of primes p up to x, for which the reduction of a non-CM elliptic curve E/Q at p is supersingular. Though the conjecture is still open, we now have unconditional upper and lower bounds thanks to the work of several mathematicians in the past few decades. However, less has been studied for the distribution of supersingular primes for abelian surfaces (even conjecturally). In this talk, I will present my recent work on unconditional upper bounds for the number of primes p up to x, for which the reduction of a fixed abelian surface at p is supersingular.

IV. Speaker: **Yotam Hendel **(KU Leuven)

Title:* On uniform dimension growth bounds for rational points on algebraic varieties*

Abstract: Given an integral projective variety X defined over Q of degree at least 2, the dimension growth conjecture, now a theorem following works of Browning, Heath-Brown and Salberger, provides an upper bound on the number of rational points of bounded height lying on X. A stronger variant of this conjecture, which is still open for cubics, furthermore predicts uniformity of these bounds when fixing the degree of X and the dimension of its ambient space.

In this talk, I will report on current developments which go beyond the classical uniform dimension growth bounds, focusing on an affine variant (which implies the projective one). This is based on recent work in the case of affine hypersurfaces, joint with Cluckers, Dèbes, Nguyen and Vermeulen.

### January 23rd 2024, Leiden

I. Speaker: **Robin de Jong** (Leiden)

Title: *Introduction to Arakelov theory*

II. Speaker: **JanVonk **(Leiden)

Title: *Blessings and curses in rational points on curves*

III. Speaker: **Ronald van Luijk **(Leiden)

Title: *Bounding Picard numbers*

IV. Magma demonstration: **Martin Bright **(Leiden)

### December 8th 2023, Groningen

I. Speaker: **Timo Keller** (Groningen)

Title: *Survey talk on the conjecture of Birch and Swinnerton-Dyer*

II. Speaker:** Chia-Fu Yu** (Academia Sinica)

Title: *Survey talk on Shimura varieties *

III. Speaker: **Dino Festi** (Padova)

Title: *Black holes and rationalizations*

Abstract: Physicists interested in high energy physics often encounter Feynman integrals presenting square roots in their argument. Exact solutions of these integrals are normally out of reach and so they are usually solved numerically. In order to achieve higher precision in the numeric evaluation, it is necessary to find a change of the variables of the integral that makes the square root disappear. Deciding the existence of such a change of variable is an algebraic problem that can be naturally translated into investigating the unirationality of a variety. The original problem can be generalized to sets of square roots. We will present a case coming from the study of two black holes, treated also using modularity results. The content of this talk is joint work with Bert van Geemen.

IV. Speaker: **Sara Mehidi** (Utrecht)

Title: *Extending torsors over regular models of curves*

Abstract: We give here an approach of the problem of extending torsors defined on the generic fiber of a family of curves. The question is to extend each of the structural group and the total space of the torsor above the family. The origin of this problem goes back to the work of Grothendieck who solved (positively) the case of a constant group of order prime to the residual characteristic. When we are interested in algebraic varieties from an arithmetic point of view, it is natural to consider torsors under a finite flat group that is not necessarily constant: we talk about fppf torsors. In fact, we already know from the literature that an fppf extension of the torsor does not always exist. So the idea is to look for a solution in a larger category, namely the category of logarithmic torsors. After providing some necessary background on log geometry, we first show that the existence of such an extension amounts to extending group functors and morphisms between them. We then try to understand the obstruction for the extended log torsor to come from an fppf one.

### November 8th 2023, Utrecht

I. Speaker: **Marta Pieropan** (Utrecht)

Title: *Introduction to Campana Points*

II. Speaker:** Valentijn Karemaker** (Utrecht)

Title: *Introduction to Lifting*

III. Speaker: **Martin Lütke** (Groningen)

Title: *Rational points and the étale fundamental group*

Abstract: According to Grothendieck's anabelian philosophy, the étale fundamental group of a variety contains a wealth of arithmetic information. In particular, the Section Conjecture predicts that the set of rational points of a hyperbolic curve can be described purely in terms of the étale fundamental group. Recently, Minhyong Kim used a unipotent version of the étale fundamental group to develop a non-abelian generalisation of the Chabauty method which can be used to effectively compute the set of rational points in some cases. I shall explain these ideas, discuss the example of the thrice-punctured line, and mention recent work with Alex Betts and Theresa Kumpitsch exploring the connection between the Section Conjecture and the Chabauty-Kim method.

IV. Speaker: **Sergio Troncoso Igua** (Universidad Técnica Federico Santa María)

Title: *Elliptic fibrations on K3 surfaces admitting strictly elliptic involution*

Abstract: In this talk, we will address the classification of elliptic fibrations X --> P^1 of K3 surfaces X, which admit a non-symplectic involution i, which fixes only a smooth curve C_g of genus g>1. The quotient X/i=Z is a smooth rational surface. In our case of interest, the quotient surface is a Del Pezzo surface. Using the characterization of the conical bundles on Del Pezzo surfaces we will describe the genus zero fibrations in the pairs (X,i). Furthermore, we will show the type of singular elliptical fibers that are theoretically admissible and which ones are realizable. Joint work with P. Comparin,P. Montero, and Y. Prieto.

### October 9th 2023, Leiden

I. Speaker: **Soumya Sankar **(Utrecht)

Title: *Counting rational points on stacks*

Abstract: Stacks are ubiquitous in algebraic geometry and often come up naturally in the context of moduli problems. In recent years, there has been increased interest in studying the arithmetic of stacks, and using their structure to answer more classical questions in number theory. In this talk, I will give a brief introduction to rational points on stacks, discuss height functions on them, and describe some applications to counting elliptic curves with prescribed level structure and bounded height.

II. Speaker: **Haowen Zhang **(Leiden)

Title: *Strong and weak approximation problems of homogeneous spaces of algebraic groups over certain function fields
*Abstract: Strong and weak approximation properties of algebraic varieties generalise the Chinese Remainder Theorem in a natural way. Such problems have been studied for homogeneous spaces of algebraic groups over number fields, and the Brauer-Manin obstruction played an important role. I present the work of my PhD dissertation where I studied such problems over some two-dimensional geometric function fields, and also over p-adic function fields. Analogous notions of the Brauer-Manin obstruction can be defined (the so-called “reciprocity obstruction” over p-adic function fields) to tackle such problems.

III. Speaker: **Ronald van Luijk** (Leiden)

Title: *del Pezzo surfaces of degree 1*

### September 6th 2023, Groningen

(Introductory session)

## Learning Seminars

## Conferences

We are organizing an **instructional workshop** on rational points, aimed at junior researchers, in Groningen from 04-11-2024 to 08-11-2024.