Curves over finite fields and arithmetic of K3 surfaces

A conference on the occasion of Jaap Top's 60th 62nd Birthday

August 30 - September 1, 2022.

Bernoulli Institute, Rijksuniversiteit Groningen



In case you are interested in attending this workshop please register here by August 20. We have limited financial support for young particpants. Please register by August 1 and send an email to one of the organizers if you want to apply for support.


Tuesday August 30:
11.00-12.00 Ben MoonenThe Tate Conjecture for Gushel-Mukai varieties
13.15-14.15 Nurdagül Anbar Meidl (online)On nilpotent automorphism groups of curves
14.45-15.45 Maciej Ulas (online)On certain Diophantine equations with functions counting partitions
16.00-17.00 Ronald van LuijkAutomorphism groups of K3 surfaces over fields that are not algebraically closed
Wednesday August 31:
9.30-10.30 Marius van der PutStokes, Painlevé and quantum differential equations
11.00-12.00 Florian Hess Limits of recursive towers
13.15-14.15 Bert van Geemen (online) Contractions of hyper-Kähler fourfolds and the Brauer group
14.45-15.45 Masato Kuwata Finding points defined over cyclic sextic extensions of an elliptic curve using a K3 surface
16.00-17.00 Elisa Lorenzo GarcíaLower bounds on the maximal number of rational points on curves over finite fields
Thursday September 1 (Intercity Number Theory Seminar):
11.30-12.30 Matthias SchüttExplicit RM for K3 surfaces
13.30-14.30 Cecília SalgadoNon-thin rank jumps for elliptic K3 surfaces
15.00-16.00 Tim DokchitserModels of hyperelliptic curves
16.15-17.15 Irene Bouw Computing Weil representations of superelliptic curves
Dinner (please register)

All talks will take place in room 105 of the Bernoulliborg.

Titles and Abstracts

Titles and Abstracts

On nilpotent automorphism groups of curves - Nurdagül Anbar

Let \(C\) be an absolutely irreducible non-singular projective curve of genus \(g \geq 2\) defined over an algebraically closed field of positive characteristic \(p\). In this talk, we give a new result on the automorphisms of \(C\). More precisely, we show that the order of a nilpotent subgroup \(G\) of its automorphism group is bounded by \(16(g-1)\) when \(G\) is not a p-group. We observe that if \(|G| = 16(g-1)\), then \((g-1)\) is a power of 2. Furthermore, we provide an infinite family of curves attaining the bound.

This is a joint work with Burçin Günes.

Computing Weil representations of superelliptic curves - Irene Bouw

Superelliptic curves are curves \(Y\) that admit a map \(f\colon Y \rightarrow \mathbb{P}^1\) that becomes cyclic after an extension of scalars. We consider superelliptic curves defined over a \(p\)-adic field \(K\) with potentially good reduction to characteristic \(p\). In an earlier paper we determined a Galois extension \(L/K\) together with a smooth model of \(Y\) over \(L\). Building on work of Dokchitser-Dokchitser, we explain how to compute the Weil representation of \(Y\) from the reduction of \(Y\). A key step is point counting on suitable twists of the reduction of \(Y\) in characteristic \(p\). This is joint work with with Do and Wewers.

Models of hyperelliptic curves - Tim Dokchitser

It is an important question how to find a regular model of a curve C/K with respect to some valuation on K. Motivated by the Birch-Swinnerton-Dyer conjecture, in 1972 Tate gave an algorithm for elliptic curves, based on the Kodaira-Neron classification ("Tate's algorithm"). Later, Liu gave an algorithm in genus 2, based on the Namikawa-Ueno classification. There have been a lot of recent activity to extend these results to higher genus, and in June 2022 Simone Muselli gave a general algorithm for hyperelliptic curves of arbitrary genus in residue characteristic<>2. It is very much in the spirit of Tate's algorithm, and is also as explicit, uses minimal machinery, and works over any discrete valuation ring. In this talk, I will give an overview of his work and some related results.

Contractions of hyper-Kähler fourfolds and the Brauer group - Bert van Geemen

The exceptional locus of a birational contraction on a hyper-Kähler fourfold of \(K3^{[2]}\)-type is a conic bundle over a K3 surface. These conic bundles are projectivized (twisted) rank two vector bundles. We discuss the associated Mukai vectors, Brauer classes (B-fields) and Heegner divisors. We also give various examples of such conic bundles.

Limits of recursive towers - Florian Hess

Recursive towers provide means for explicit constructions of curves over finite fields with asymptotically many rational points in comparison to their genus, as quantified by their limit. The limit is a lower bound for Ihara's constant and is often not known by itself, but only bounded from below by a known constant. We discuss some new results by which limits can be precisely determined in many cases. This is work of Dietrich Kuhn and myself.

Finding points defined over cyclic sextic extensions of an elliptic curve using a K3 surface - Masato Kuwata

Let \(E\) be an elliptic curve over \(\mathbb{Q}\). We consider the problem of finding points on \(E\) defined over some cyclic extension of fixed degree \(d\). The existence of such a point is linked, conjecturally, to the vanishing of the twist of the \(L\)-function of \(E\) by a Dirichlet character. A geometric approach to this problem is to translate it to finding \(\mathbb{Q}\)-rational points on a generalized Kummer variety defined as the quotient of \(E^{d-1}\) by a cyclic group action. In this talk we give a new positive result for \(d=6\). In this case the related Kummer variety is a K3 surface with an elliptic fibration. We also discuss the case \(d=5\) and \(8\), and explain the pessimistic perspective. This is a joint work with Hershy Kisilevsky.

Lower bounds on the maximal number of rational points on curves over finite fields - Elisa Lorenzo García

For a long time people have being interested in finding and constructing curves with many points. For genus 1 and genus 2 curves, we know how to construct curves over any finite field of defect less than 1 or 3 (respectively), i.e. with a number of points at distance at most 1 or 3 to the upper bound given by the Hasse-Weil-Serre bound. The case of genus 3 is still open after more than 40 years of research. In this talk I will take a different approach based on the random matrix theory of Katz-Sarnak to prove the existence for all \(\epsilon>0\) of curves of genus \(g\) over Fq with more than \(1+q+(2g-\epsilon)\sqrt{q}\) points for \(q\) big enough. I will also discuss some explicit constructions as well as some consequences to the Serre obstruction problem (an asymmetric behaviour of the distribution of the trace of the Frobenius for curves of genus 3).

This is a joint work with J. Bergström, E. Howe and C. Ritzenthaler.

Automorphism groups of K3 surfaces over fields that are not algebraically closed - Ronald van Luijk

A consequence of the Torelli Theorem for K3 surfaces is that the automorphism group of a complex algebraic K3 surface is finite if and only if the automorphism group of its Picard lattice modulo the Weyl group of that lattice is finite. In other words, whether or not the K3 surface has finite automorphism group depends only on its Picard lattice. Over general fields this is no longer true. We will give examples of what can go wrong, and we also give some analogs of finiteness statements that are still correct. This is joint work with Martin Bright and Adam Logan.

The Tate Conjecture for Gushel-Mukai varieties - Ben Moonen

I will report on joint work with Lie Fu, in which we prove the Tate Conjecture for even dimensional Gushel-Mukai varieties over fields of characteristic p>3. These Gushel-Mukai varieties have a motive of K3 type. Over the complex numbers they have been studied extensively, and they turn out to have beautiful connections to hyperkähler geometry. For expository reasons, I will focus on the case of GM 6-folds. Our approach to the Tate Conjecture is modelled after Madapusi Pera's proof of the TC for K3 surfaces. To make this work we need a number of basic properties of GM 6-folds, such as the fact that they have no nonzero global vector fields. This turns out to be a hard nut that we were able to crack only with some help from Magma.

Stokes, Painlevé and quantum differential equations - Marius van der Put

Jaap Top and I have a long history of joint work. We have published, among other things, on the Painlevé equations I, II, III, IV, degV, V.

Our present work also concerns quantum differential equations. It is interesting to mention that some of our collaboration on these topics is with three Colombian mathematicians, namely Primitivo Acosto Humánez, Camilo Sanabria Malagón and Alex Cruz Morales.

In this talk we will explain the ``Stokes phenomenon'', give a short introduction to Painlevé equations and try to say something about quantum differential equations. Finally, some examples will show the computational aspect of the subject.

Non-thin rank jumps for elliptic K3 surfaces - Cecília Salgado

We discuss recent progress on the variation of the Mordell-Weil rank in families of elliptic curves over number fields. In the case of elliptic K3 surfaces, we show, under certain conditions, that the set of fibres for which the Mordell-Weil rank is strictly larger than the generic rank is not thin, as a subset of the base of the fibration. This is based on joint work with Hector Pasten.

Explicit RM for K3 surfaces - Matthias Schütt

While the CM case is fairly well understood, the RM case (with a field different from Q), remains quite mysterious, with no explicit examples exceeding one-dimensional families.

I will report on joint work in progress with Bert van Geemen which aims to push our knowledge much further. One of our two main approaches is partly based on work of our birthday boy.

On certain Diophantine equations with functions counting partitions - Maciej Ulas

Let \(\mathbb{N}\) be the set of non-negative integers and \(\mathbb{N}_{+}\) the set of positive integers. Let \(A\subset\mathbb{N}_{+}\) and take \(n\in\mathbb{N}\). By an \(A\)-partition \(\lambda=(\lambda_{1},\ldots, \lambda_{k})\), of a non-negative integer \(n\) with parts in \(A\), we mean a representation of \(n\) in the form $$ n=\lambda_{1}+\ldots+\lambda_{k}, $$ where \(\lambda_{i}\in A\). The representations of \(n\) differing only in order of the terms are counted as the same. Let \(P_{A}(n)\) denote the number of \(A\)-partitions of \(n\). For given \(A\subset\mathbb{N}_{+}\) we consider the problem of finding integer solutions of the Diophantine equation $$ P_{A}(x)=f(x_{1},\ldots,x_{k}), $$ where \(f\in\mathbb{Z}[x_{1},\ldots, x_{k}]\). In the first part of the talk, I will report on a joint results with Szabolcs Tengely (University of Debrecen, Hungary) concerning the Diophantine equation $$ P_{A}(x)=P_{B}(y) $$ for different finite sets \(A, B\). In the second part, I will present recent results from joint work with Bartosz Sobolewski (Jagiellonian University, Poland) and get a full characterization (in \(n\)) of the solutions of the Diophantine equation $$ P_{A}(n)=x^2+y^2+z^2, $$ for the set \(A=\{2^{i}:\;i\in\mathbb{N}\}\). According to our best knowledge, this is the first example of an unrestricted partition function with an infinite set \(A\) for which the characterization of values represented by a sum of three squares is obtained. I also present some results concerning representations of values of the \(m\)-th order convolution of \(P_{A}\) as sums of three squares, for \(m\) of the form \(2^{k}-1\).