Arithmetic geometry of K3 surfaces

Workshop organised by Anna Cadoret, Alexei Skorobogatov, Domenico Valloni, and Tony Várilly-Alvarado at the Bernoulli Center, EPFL, Lausanne (5 May - 9 May 2025).

For titles and abstracts see below.

Monday

9:30-10:30 Ben Moonen

coffee

11:00-12:00 Emiliano Ambrosi

lunch

2-3 Emma Brakkee

3:30-4:30 Marco D’Addezio

4:45-5:30 Livia Grammatica

5:30-… apéro

Tuesday

9:30-10:30 Kiran Kedlaya

coffee

11:00-12:00 Daniel Bragg

lunch

2-3 Ziquan Yang

3:30-4:30 Adam Morgan

4:45-5:30 Salvatore Floccari

Wednesday

9:30-10:30 Max Lieblich

coffee

11-12 Yuya Matsumoto

lunch

Thursday

9:30-10:30 Stefan Schröer

coffee

11-12 Chun-Yin Hui

lunch

2-3 Sarah Frei

3:30-4:30 Zhenghui Li

4:45-5:30 Emre Sertöz

5.30-… apéro

Friday

9:30-10:30 Matthias Schütt

coffee

11-12 François Charles

lunch

Emiliano Ambrosi

An Artin-Mumford criterion for conic bundles in characteristic 2

In the 70s, Artin and Mumford proved an irrationality criterion for a conic bundle over a field of characteristic different from 2. In a joint work with Giuseppe Ancona, we prove an analogous result in characteristic 2 and use it to deduce irrationality results in characteristic 0.

Daniel Bragg

Compactifications of supersingular twistor spaces

I will describe a geometric construction that associates to a supersingular K3 surface a collection of families of supersingular K3 surfaces over P¹. These families are obtained as moduli spaces of sheaves over certain algebras of twisted differential operators, whose construction relies on some special features of symplectic geometry in positive characteristic. I will also explain some applications to the Tate conjecture for supersingular varieties.

Emma Brakkee

Bounding Brauer groups of K3 surfaces using moduli spaces

The Brauer group of an algebraic variety is a group with many applications, in particular to the study of rational points. For a K3 surface over a number field, the transcendental part of its Brauer group is finite. It was shown by Cadoret-Charles that the size of its p-primary torsion is uniformly bounded for K3 surfaces in one-dimensional families. We give a different proof of this result for one-dimensional families of K3 surfaces with a polarization by a fixed lattice. To be precise, we construct moduli spaces of K3 surfaces with a lattice polarization and a Brauer class, and use the geometry of their complex points to prove boundedness of Brauer groups for the K3 surfaces they parametrize. I will explain the construction and give a sketch of the proof of our boundedness result. This is joint work in progress with D. Bragg and T. Várilly-Alvarado.

François Charles

Positivity of line bundles and Grauert tubes in arithmetic geometry

I will discuss classical and less classical results regarding positivity conditions for Hermitian line bundles with singular metrics. I will focus on the analogy with classical results in complex geometry, while discussing new methods using infinite-dimensional geometry of numbers. This is joint work with Jean-Benoît Bost.

Marco D’Addezio

The Hecke orbit conjecture

I will talk about the proof of the Hecke orbit conjecture for Shimura varieties, as formulated by Chai and Oort. After reviewing the statement in the case of the moduli space of abelian varieties, I will explain the linearisation of the problem using generalised Serre-Tate coordinates. I will then present the role of monodromy groups of F-isocrystals in this context. Finally, I will say some words on how the conjecture was applied by Bragg and Yang to establish a criterion of potential good reduction for K3 surfaces. This is joint work with Pol van Hoften.

Salvatore Floccari

On some families of K3 surfaces of general Picard rank 16

I will discuss some 4-dimensional families of complex K3 surfaces which are related to abelian fourfolds of Weil type with discriminant 1 via the Kuga-Satake construction. I will explain how these K3 surfaces are also naturally associated with certain 4-dimensional families of the six-dimensional hyper-Kähler varieties discovered by O'Grady. We use this to prove that the Kuga-Satake correspondence is algebraic for any of these K3 surfaces. We further obtain a new proof of Markman's theorem that the Hodge conjecture holds for any Weil fourfold with discriminant 1. As applications, we establish the conjectures of Hodge and Tate for many hyper-Kähler varieties of OG6-type and their powers. The results presented are joint work with Lie Fu.

Sarah Frei

Relative Severi-Brauer varieties on K3 surfaces via hyperkahler manifolds

The Brauer group of a surface is an essential tool in the study of the arithmetic of surfaces over non-closed fields, but it can also bear on other geometric problems, such as the rationality of fourfolds (even over the complex numbers). Geometric realizations of Brauer classes on surfaces, as étale projective bundles, often play a key role in our understanding of such applications. In this talk, I will discuss joint work with Jack Petok and Anthony Várilly-Alvarado, in which we consider constructions of these étale projective bundles for Brauer classes on K3 surfaces. This builds on earlier results and predictions of Hassett and Tschinkel, as well as on recent results of van Geemen and Kaputska, who show that some (but not all) 2-torsion Brauer classes on K3 surfaces have realizations as the exceptional locus of a divisorial contraction on a hyperkahler fourfold.    

Livia Grammatica

Fppf cohomology in positive characteristic via the Nygaard filtration

Flat p-adic cohomology of varieties in positive characteristic has classically been studied via differential techniques such as the de Rham-Witt complex. In this talk, we show how many of the results that Illusie obtains in his first de Rham-Witt paper can be proved using the elementary properties of the Nygaard filtration. As a further application we compute the action of multiplication-by-n on the fppf cohomology of an abelian variety.

Chun-Yin Hui

On distribution of supersingular primes of abelian varieties and K3 surfaces

Let K be a global field and let X/K be a non-CM abelian variety or non-CM K3 surface. We prove that the density of the supersingular primes of X is zero. When K is a number field, we obtain asymptotic upper bounds of the counting function for supersingular primes by applying the effective Chebotarev density theorem for infinite Galois extensions due to Serre.

Kiran Kedlaya

The Honda-Tate problem for K3 surfaces

The zeta function of a K3 surface over a finite field F_q includes a single factor that is not determined solely by q, which records the Frobenius on primitive middle etale cohomology. If we view this factor as a function on F_q-points of the moduli space of K3 surfaces, the Honda-Tate problem is to determine the image of this function. We survey the state of knowledge around this problem, including recent and ongoing computational work.

Zhenghui Li

Boundedness of the p-primary torsion of geometric Brauer groups 

Let X be a smooth projective integral variety over a finitely generated field k of characteristic p>0. We show that the finiteness of the exponent of the p-primary part of Br(X_{k^s})^{G_k} is equivalent to the Tate conjecture for divisors, generalizing D'Addezio's theorem for abelian varieties to arbitrary smooth projective varieties. In combination with the Leray spectral sequence for rigid cohomology derived from the Berthelot conjecture recently proved by Ertl-Vezzani, we show that the cokernel of Br_nr(K(X)/F_p) \to Br(X_{k^s})^{G_k} has finite exponent. This completes the p-primary part of the generalization of the Artin-Grothendieck theorem on the relation between Brauer groups and Tate-Shafarevich groups to higher relative dimensions. This is joint with Yanshuai Qin.

Max Lieblich

Some questions about Brauer groups and K3 surfaces

I will review some basic results about splitting Brauer classes on various varieties. Then I will discuss the somewhat perplexing case of K3 surfaces. Finally, I will spend some time fantasizing about how one might try to bound the size of the Brauer group of a K3 surface over a finite field. This talk will have a lot more questions than answers.

Yuya Matsumoto

Supersingular abelian surfaces in characteristic 2 and inseparable Kummer surfaces

Unlike in the case of all other abelian surfaces, Kummer surfaces attached to supersingular abelian surfaces in characteristic 2 are not K3 surfaces but rational surfaces. We discuss our attempt in progress to attach K3 surfaces to such abelian surfaces. These K3 surfaces are “limits” of usual Kummer K3 surfaces, and are certain kind of supersingular K3 surfaces, which we name inseparable Kummer surfaces.

Ben Moonen

Computing discrete invariants of varieties in positive characteristic

Here is a motivating question, which is a special case of a more general problem: The moduli space of K3 surfaces in characteristic p is stratified by the height of the formal Brauer group, and the smallest stratum (the supersingular locus) is further stratified by the Artin invariant. If we give ourselves an explicit K3 surface, e.g. a quartic surface in P³, how can we calculate in which stratum it lies? In my talk, I will explain what an F-zip is (I will not assume you already know this), and how this relates to the above problem. Under mild technical assumptions, we can associate an F-zip to every smooth projective variety in characteristic p, and such F-zips have been classified. I will explain some new techniques that allow us to calculate the F-zips of some accessible types of varieties, such as projective hypersurfaces.

Adam Morgan

On the Hasse principle for degree 4 del Pezzo surfaces 

I will discuss recent work with Skorobogatov, and work in progress with Lyczak, establishing the Hasse principle for classes of degree 4 del Pezzo surfaces, conditional on finiteness of certain Tate-Shafarevich groups. Key to the method is the study of an auxiliary family of Kummer surfaces. 

Stefan Schröer

K3 surfaces over small number fields and Kummer construction in families

We construct K3 surfaces over certain S₃-number fields that have good reduction everywhere. To this end we develop a theory of Kummer constructions in families, based on Romagny's notion of the effective models, here applied to sign involutions. This includes quotients of non-normal surfaces by infinitesimal group schemes in characteristic two, as developed by Kondo and myself. By the results of Brieskorn and Artin, the resulting families of normal K3 surfaces admit simultaneous resolutions of singularities, at least after suitable base changes. These resolutions are constructed in two ways: First, by blowing-up families of one-dimensional centers that acquire embedded components. Second, by computing various l-adic local systems in terms of representation theory and invoking Shepherd-Barron’s results on the resolution functor, a highly non-separated algebraic space.

Matthias Schütt

Finite symplectic automorphism groups of supersingular K3 surfaces

Finite symplectic automorphism groups on complex K3 surfaces have famously been classified by Mukai, based on work of Nikulin. I will report on joint work with Hisanori Ohashi which extends this to positive characteristic. While the tame case retains a close connection to the Mathieu group M₂₃ (extending work of Dolgachev-Keum), we will develop a unified approach using symmetries of the Leech lattice which also covers all wild cases for superspecial K3 surfaces.

Emre Sertöz

Computing the Picard lattice of a quartic K3 as a Galois module

I will present a method, developed with Edgar Costa (MIT), that aspires to compute the Picard lattice of a smooth quartic surface in P³ as a Galois module, starting from its defining equation. Although there are constraints to guarantee the success of the method, the method always produces rigorous results despite relying on numerical approximations of periods. The method builds on the ideas that seeded IVHS in the work of Griffiths--Harris (1983) and was further developed by Movasati--Sertöz (2021) and Cifani--Pirola--Schlesinger (2022). With Edgar Costa, we further engineered the method to handle mathematically hazardous environments, chief among them the Mukai--Klein quartic. This surface contains no lines, conics, or twisted cubics, but 133056 quartic rational curves, making Galois reconstruction especially challenging. The problem was originally posed by Noam Elkies, and we incorporate many of his insights.

Ziquan Yang

A simple intrinsic proof of the Tate conjecture for K3's of finite height 

In the past decade, a major triumph for the Tate conjecture (over finite fields) is its resolution for K3 surfaces. However, all known proofs rely crucially on the Kuga-Satake construction—effectively outsourcing certain difficulties to the theory of abelian varieties. There is an alternative approach which seeks to prove the conjecture by linking it to finiteness statements in arithmetic geometry and using only the geometry of K3 surfaces. The spirit of this approach originated in Artin and Swinnerton-Dyer’s proof for elliptic K3's and has recently been revitalized by Lieblich-Maulik-Snowden and Charles through the moduli theory of (twisted) sheaves. In particular, Charles gave an intrinsic proof assuming the Picard number is at least 2. In this talk, I will explain an intrinsic proof that finite height K3's a priori have an even geometric Picard number, and hence proves one step further in the second approach. If time permits, I will also discuss speculations for the supersingular case.