Arithmetic geometry of K3 surfaces

Weekly seminar organised by Anna Cadoret, Alexei Skorobogatov, Domenico Valloni, and Tony Várilly-Alvarado at the Bernoulli Center, EPFL, Lausanne (31 March - 9 May 2025).

Week 1

Tuesday 1 April, 2 pm Organisational meeting of the weekly seminar

Wednesday 2 April, 2 pm Daniel Huybrechts 

The Tate-Shafarevich group of a polarised K3 surface

Abstract: The TS group of an elliptic (K3) surface parametrises the twists of the elliptic fibration. I will explain how to define the TS group of any complete linear system on a K3 surface and how to interpret its elements in geometric terms. (This is joint work with Dominique Mattei.)

Thursday 3 April, 11 am Daniel Huybrechts 

The period-index problem for unramified Brauer classes

Abstract: The period-index conjecture predicts that the index of a Bauer class can be bounded by a uniform (only depending on the dimension) power of its period. We prove that a uniform exponent can be found that only depends on the degree of the variety. (This is joint work with Dominique Mattei.)

On Thursday afternoon and Friday, the following event takes place at the conference room of the Bernoulli Center:

Basel-Dijon-EPFL-Neuchâtel-Zürich meeting

Week 2

Tuesday 8 April, 2 pm Domenico Valloni

Supersingular Brauer classes in positive characteristic

Abstract: I will talk about certain p-torsion Brauer classes in positive characteristic which can be constructed using differential forms. I will then prove that this construction yields the Brauer group of supersingular K3 surfaces over a closed field, along with their Artin invariant. In the second part of the talk, I will talk about the role of these classes in the Brauer-Manin obstruction. This will reveal novel phenomena in positive characteristic, such as the positive effect of the failing of the Bogomolov-Sommese vanishing on the Brauer-Manin obstruction.

Wednesday 9 April, 2 pm Yuan Yang

Unipotent torsion in the Brauer group

Abstract: Let X be a proper smooth variety over an algebraically closed field k of characteristic p. The classical logarithmic de Rham-Witt theory shows that the Brauer group Br(X) contains a unipotent subgroup. We show that this unipotent group U is trivial for ordinary varieties, Enriques surfaces, Godeaux surfaces, and (quasi)hyperelliptic surfaces. Under some conditions, using work of Ekedahl and Crew, we express the dimension of U in terms of the Hodge numbers and the Newton polygon of X. In a joint work in progress with Skorobogatov and Zarhin, we give partial results on the isogeny class of the unipotent group U for abelian varieties: we compute the dimension of U[p] for principally polarized abelian varieties from its Ekedahl-Oort type. Using this, we compute the isogeny class of U for principally polarized abelian 3-folds. 

Week 3

Tuesday 15 April, 2 pm Eva Bayer-Fluckiger

Automorphisms of K3 surfaces and cyclotomic polynomials

Abstract: Let X be a complex projective K3 surface and let T_X be its transcendental lattice. The characteristic polynomials of the isomorphisms of T_X induced by automorphisms of X are powers of cyclotomic polynomials. Which powers of cyclotomic polynomials occur? The aim of this talk is to answer this question, as well as related ones.

Wednesday 16 April, 2 pm Emma Brakkee

Moduli spaces of K3 surfaces with a Brauer class

Abstract: Twisted K3 surfaces are pairs of a K3 surface together with a Brauer class. They are parametrized by moduli spaces of twisted K3 surfaces of fixed order with an ample line bundle. Over the complex numbers, one can construct these moduli spaces using Hodge theory, which we did in previous work to study the relation with cubic fourfolds. In a current joint project with D. Bragg and A. Várilly-Alvarado, we generalize the above, constructing moduli spaces of twisted K3 surfaces with a polarization by a lattice of higher rank. We use them to study the size of the transcendental parts of Brauer groups of K3 surfaces over number fields. In this talk, I will focus on the construction of these moduli spaces and what they look like over the complex numbers.

Week 4

Tuesday 22 April, 2 pm Kazuhiro Ito

Deformation theory of K3 surfaces via prismatic cohomology

Abstract: Let X be a K3 surface over a perfect field k of positive characteristic p. I will prove that formal deformations of X over a complete discrete valuation ring O_K of mixed characteristic with residue field k are classified by Breuil-Kisin modules of K3 type via prismatic cohomology. Along the way, I will show that in characteristic 2, the cup product pairing of de Rham cohomology of any formal deformation of X comes from a quadratic form. As an application, combined with Ogus' crystalline Torelli theorem and a result of Bhatt-Scholze, I will give a characterization of lattices in crystalline representations of K which arise as the p-adic étale cohomology of the (rigid-analytic) generic fiber of a p-adic formal K3 surface over O_K when k is algebraically closed and p is odd.

Thursday 24 April, 2:15-4 pm John Voight

Modularity of K3 surfaces

Abstract: In the introductory first part, we give an overview about modularity results for K3 surfaces.  In the second part, we report on joint work with Edgar Costa, Stephan Elsenhans, and Jorg Jahnel concerning explicit modularity of K3 surfaces with complex multiplication of large degree.  (30 minutes general talk followed by 45 minutes research talk)

Week 5

Wednesday 30 April, 2 pm Gregorio Baldi

The Noether-Lefschetz locus and the questions of Harris and Voisin

Abstract: After recalling the classical results on the distribution of the components of the NL locus from the 80s and early 90s, I will explain how they relate to the ‘completed Zilber-Pink philosophy’ developed with B. Klingler and E. Ullmo. Armed with this viewpoint, we will show that the so called exceptional components are not Zariski dense in the moduli space of smooth surfaces of degree d, for d>4. The novelty is that they, in some sense, are explained by the presence of extra ‘higher Hodge tensors’.

Thursday 1 May, 2pm Damián Gvirtz-Chen

Hilbert modular surfaces and the Inverse Galois Problem

Abstract: Let K be a real quadratic field and let X be the associated Hilbert modular surface (at base level) over Q. If X is a K3 surface, we show that it satisfies the Hilbert Property. As a consequence, we can prove new instances of the Regular Inverse Galois Problem for simple groups of type PSL_2(F_{p^2}) where p satisfies certain congruence conditions. Joint work with J. Demeio.

Thursday 1 May, 3:30 pm Kazuhiro Ito

Kuga-Satake construction in mixed characteristic and its applications

Abstract: I will discuss the Kuga-Satake construction, with a particular emphasis on its relationship with (integral) p-adic Hodge theory. As applications, I will outline how the construction leads to results on CM liftings of K3 surfaces over finite fields, as well as the Tate conjecture and the (Hodge) standard conjecture for their self-products. This is based on joint work with Tetsushi Ito and Teruhisa Koshikawa. If time permits, I will make a few comments on (a part of) an on-going project with Tetsushi Ito, Teruhisa Koshikawa, Teppei Takamatsu, and Haitao Zou, concerning a Kuga-Satake type construction for higher degree p-adic cohomology of hyper-Kähler varieties.