**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**

S*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.