­K3 surfaces and Galois representations

Abstracts of talks

Emiliano Ambrosi

Specialization of Néron-Severi groups in positive characteristic slides

Let $k$ be an infinite finitely generated field of characteristic $p>0$. We fix a smooth separated geometrically connected scheme $X$ of finite type over $k$ with generic point $\eta$ and a smooth proper morphism $f:Y\rightarrow X$. In this talk we prove that there are "lots" of $x\in X$ such that the fibre of $f$ at $x$ has the same geometric Picard rank as the generic fibre. In characteristic zero, this has been proved using Hodge theoretic methods. To extend the argument in positive characteristic we use the comparison between different p-adic cohomology theories and independence techniques. We explain some applications of the result to uniform boundedness of Brauer groups, the Tate conjecture, abelian varieties, and to proper families of projective varieties.

Gregorio Baldi

Local to global principle for the moduli space of K3 surfaces slides

Recently Patrikis, Voloch and Zarhin have studied the finite descent obstruction for the moduli space of principally polarised abelian varieties. We show an analogous result for K3 surfaces, under some technical restrictions. Our approach is possible since abelian varieties and K3s are quite well described by `Hodge-theoretical' results. In particular the theorem we present can be interpreted as follows: assuming the Tate, Fontaine-Mazur and Hodge conjectures, a weakly compatible system of ℓ-adic representations that `looks like’ the one induced by the transcendental lattice of a K3 surface (defined over a number field) determines a Hodge structure of K3 type which in turn determines a K3 surface (which may be defined over a number field).

Martin Bright

Finiteness results for K3 surfaces over arbitrary fields slides

Over an algebraically closed field, various finiteness results are known regarding the automorphism group of a K3 surface and its action on the Picard lattice. For example, whether the automorphism group is finite depends only on the abstract isomorphism type of the Picard lattice. We investigate the extent to which these results remain true over arbitrary fields, and give examples illustrating how behaviour can differ from the algebraically closed case. This is joint work with Adam Logan and Ronald van Luijk.

Anna Cadoret

On uniform boundedness of arithmetico-geometric invariants in one-dimensional families slides

Let $k$ be a finitely generated field of characteristic $p\geq 0$ and $X$ a smooth, separated and geometrically connected curve over $k$. Fix a prime $\ell$. I will describe a general uniform open image theorem for $\ell$-adic representations of the etale fundamental of $X$ (Cadoret-Tamagawa, $p=0$; Ambrosi, $p>0$). I will explain how this uniform open image theorem can be applied to obtain uniform bounds for arithmetico-geometric invariants encoded in $\ell$-adic cohomology in families of smooth proper varieties parametrized by $X$. I will discuss more specifically the $\ell$-primary torsion of abelian varieties (Cadoret-Tamagawa) and of the Galois-fixed part of the geometric Brauer group (Cadoret-Charles, $p=0$; Ambrosi, $p>0$)). I will also give a brief sketch of the proof of the uniform open image theorem. If times allows, I may discuss one more application, to the rank of abelian varieties. For the application to Brauer groups, I will limit myself to the case $p=0$. The case $p>0$, which is far much technical, is due to Ambrosi and will be discussed in his talk.

Daniel Huybrechts

Finiteness of polarized K3 surfaces

I shall explain how to deduce a consequence of the cone conjecture, namely that up to automorphisms a K3 surface only admits finitely many polarizations of fixed degree, from moduli theory. In an orthogonal directions, involving twistor spaces, I will discuss the question how many fibres can be isomorphic to each other. Partial answers are presented for the case of K3 surfaces with CM.

Evis Ieronymou

Zero-cycles on K3 surfaces over number fields slides

Let X be a smooth, projective, geometrically connected variety over a number field. It is conjectured that if there is a family of local zero-cycles of degree one orthogonal to the Brauer group of X then there exists a zero-cycle of degree one on X. The analogous statement for the existence of rational points is known to fail in general. However there is some evidence to suggest that for K3 surfaces it might be true, i.e. that the Brauer-Manin obstruction to the Hasse principle is the only one for K3 surfaces. In this talk we provide a link between the two by showing that if the Brauer-Manin obstruction to the Hasse principle is the only one for K3 surfaces (over any number field), then the Brauer-Manin obstruction to the existence of a zero-cycle of degree 1 is the only one for K3 surfaces. The proof is based on an idea of Liang to use the trivial fibration over the projective line.

Jörg Jahnel

Real and complex multiplication on K3 surfaces via period integration slides

I report on a new approach, as well as some related experiments, to construct families of K3 surfaces having real or complex multiplication. Fundamental ideas include considering the period space of marked K3 surfaces, determining the periods by numerical integration, as well as tracing the modular curve by a numerical continuation method. This is joint work with Andreas-Stephan Elsenhans (Paderborn).

Ben Moonen

Images of Galois representations and the Mumford-Tate conjecture slides

I will report on joint work with Anna Cadoret. Let X be a smooth projective over a finitely generated field k of characteristic 0, and let H be the l-adic cohomology of X in some given degree. The Galois group of k acts on H and the Mumford-Tate conjecture predicts that the image of this Galois representation is Zariski dense in the Mumford-Tate group over Q_l. This conjecture gives an important link between Hodge theory and the theory of Galois representations. Serre has conjectured that in fact something much finer should be true, concerning the actual image of the Galois representation and not only its Zariski closure. For abelian varieties we prove that the Mumford-Tate conjecture implies these finer results. For K3 surfaces the situation is even better, and we obtain an unconditional result about the image of the Galois representation.

Masahiro Nakahara

Index of fibrations and Brauer classes that never obstruct the Hasse principle slides

We study the classes in the Brauer group of varieties that never obstruct the Hasse principle. We prove that for a variety with a genus 1 fibration, if the generic fiber has a zero-cycle of degree d over the generic point, then the Brauer classes whose orders are prime to d do not play a role in the Brauer-Manin obstruction. As a result we show that the odd torsion Brauer classes never obstruct the Hasse principle for del Pezzo surfaces of degree 2, certain K3 surfaces, and Kummer varieties.

Otto Overkamp

Kummer surfaces and Galois representations

Kummer surfaces are K3 surfaces which can be obtained from Abelian surfaces using a simple geometric construction. I shall discuss an analogue of the Néron-Ogg-Shafarevich criterion for Kummer surfaces, and explain how semistable reduction of Abelian surfaces and their associated Kummer surfaces are related. Time permitting, I shall say a few words about Kulikov models.

Matthias Schütt

Singular K3 surfaces of class number 2 slides

Singular K3 surfaces, i.e. those with Picard number 20, carry a rich arithmetic structure much of which is captured by the transcendental lattice. Its discriminant gives rise to the class number of the K3 surface. In this talk I will report on an ongoing project, in a bigger framework, to determine which singular K3 surfaces are defined over Q (in parallel with CM elliptic curves). As a first step, I will derive the affirmative solution for class number 2.

Lenny Taelman

Ordinary K3 surfaces over finite fields

We give a description of the category of ordinary K3 surfaces over a finite field in terms of linear algebra data over Z. This gives an analogue for K3 surfaces of Deligne's description of the category of ordinary abelian varieties over a finite field, and refines earlier work by N.O. Nygaard and J.-D. Yu. Two important ingredients in the proof are integral p-adic Hodge theory, and a description of CM points on Shimura stacks in terms of associated Galois representations. References: arXiv:1711.09225, arXiv:1707.01236.

Domenico Valloni

K3 surfaces and complex multiplication slides

The classical theory of complex multiplication for abelian varieties could be considered as an intersection point between Hodge theory, arithmetic geometry and algebraic number theory. For example, starting with a CM elliptic curve over the complex  numbers, the theory allows us to descend this geometric object to a precise number field (the Hilbert class field of the CM field) and obtain detailed information about the associated Galois representation, via class field theory. In retrospect, this is possible  because the canonical model of the Siegel Shimura variety is also a solution to the naturally associated moduli problem. One then expects that similar results must hold for K3 surfaces, if not because also in this case the canonical model of K3 Shimura varieties  is closely related (via the Period morphism) to the moduli spaces of principally polarised K3 surfaces. In this talk, we present our results to this direction, showing also how the CM theory for K3 surfaces is a natural tool to perform concrete computations  on their Brauer groups, fields of moduli and fields of definition.

Tony Várilly-Alvarado

Cubic fourfold of discriminant 18 and odd-torsion Brauer-Manin obstructions to the Hasse Principle on general K3 surfaces slides

In 2014, Skorobogatov and Ieronymou asked if odd-torsion classes in the Brauer group of a K3 surface over a number field could obstruct the existence of rational points. Corn and Nakahara answered this question affirmatively with a 3-torsion algebraic class on a K3 surface of degree 2. We show that transcendental 3-torsion classes arising from a cubic fourfold containing a sextic elliptic surface can also obstruct the existence of rational points. Our approach does not require explicit cyclic Azumaya representatives of the class; it is instead a purely geometric argument. The 3-torsion classes we construct via a geometric correspondence fit into framework that describes level structures of small level on low-degree K3 surfaces. This is joint work with Jennifer Berg.

Yuri Zarhin

uperelliptic jacobians, Brauer groups and Kummer varieties slides

In this talk we deal with jacobians $J$ of superelliptic curves $y^{\ell}=f(x)$ where $\ell$ is a prime and $f(x)$ is an irreducible polynomial over a number field $K$. It turns out that certain natural conditions on $f(x)$ and $\ell$ imply the vanishing of the $\ell$-primary component of the subgroup of Galois-invariants in the Brauer group of $J$ (over an algebraic closure of $K$). We also discuss Brauer-Manin sets of corresponding generalized Kummer varieties. This a report on a joint work with Alexei Skorobogatov.