Monday 
9:00–9:15  Welcome 
9:15–10:15  Jörg Brüdern 
Another view of Artin’s conjecture
Artin conjectured around 1930 that
a rational form of degree d in more than d^{2}
variables has nontrivial zeros in all padic fields.
This turned out to be not quite true, but has been confirmed
in important special cases, for example for diagonal forms.
We propose a diagonalisation
approach and then describe joint work with Robert
that establishes Artin’s conjecture for forms that can be
obtained by restricting a diagonal form to a hyperplane.

10:15–10:45  Coffee and discussion 
10:45–11:15  Efthymios Sofos 
The location of the primes obstructing the existence of a rational point
The first step in checking computationally whether a variety over the rationals
has a rational zero is to check whether it has a padic point for all primes p. In recent
work with Daniel Loughran we studied the typical behavior of the primes for which there is
no padic point by studying their cardinality. In upcoming work we describe the typical
behaviour of such primes by studying their location. The new input comes from Brownian
motion.

11:30–12:30  Tim Browning 
Free rational curves on hypersurfaces and the circle method
The circle method over F_{q}(t) allowed Pankaj Vishe and I to calculate the dimension and irreducibility of the moduli space of rational curves on any
smooth projective hypersurface of low enough degree. In this talk I will report on joint work with Will Sawin, in which we
go one step further and exploit a relationship with a particular system of two equations that allows us to bound the dimension of its singular locus.

12:30–14:30  Lunch and discussion 
14:30–15:30  Tony VárillyAlvarado 
Cubic fourfolds and oddtorsion Brauer–Manin obstructions on general K3 surfaces
In 2014, Skorobogatov and Ieronymou asked if oddtorsion 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 3torsion algebraic class on a K3 surface of degree 2. We show that transcendental 3torsion 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 3torsion classes we construct via a geometric correspondence fit into framework that describes level structures of small level on lowdegree K3 surfaces. This is joint work with Jennifer Berg.

15:30–16:15  Coffee and discussion 
16:15–16:45  Francesca Balestrieri 
Arithmetic of rational points and zerocycles on products of Kummer varieties and K3 surfaces
The following is joint work with Rachel Newton.
Let k be a number field. Building on results of Yongqi Liang and, more recently, of Evis
Ieronymou and
of our own, we relate, via the Brauer–Manin obstruction, the arithmetic of rational points
over finite extensions of k to that of zerocycles over k for any variety X=
Π^{n}_{i=1} X_{i} that is a finite product of Kummer varieties, K3 surfaces, and smooth
projective geometrically rationally connected varieties over k. In particular, we show
that if the Brauer–Manin obstruction is the only obstruction to the existence of rational
points on X_{i} over all finite extensions of k for each i=1,…, n, then the
Brauer–Manin obstruction is the only obstruction to the existence of a zerocycle of
degree 1 on X.

17:00–17:30  Julian Lyczak 
Brauer–Manin obstructions of order 5 on log K3 surfaces
Let X be a projective variety over Q. Manin used the Brauer group of X to develop the
Brauer–Manin obstruction, which can explain the absence of rational points. Now consider a
del
Pezzo surface X of degree 5, which always has a rational point. This implies that an
obstruction
will not exist, which also follows from the fact that the Brauer group of such a surface
is always
trivial.
The situation is completely different in the following case: we start with a del Pezzo
surface X
of degree 5 with its embedding X → P^{5}(Q).
Let D ⊆ X be a hyperplane section and consider the
complement U=X\D which is an example of a log K3 surface. If one has explicit equations
or equivalently a model over Z for the affine surface U we can study the existence of
integral
points on U using the Brauer–Manin obstruction as before. The Brauer group of such
surfaces
can contain elements of order 5 and I will show how to construct such an affine surface
over Z for
which these elements obstruct the existence of integral points.

17:30–  Dinner and free time 
Tuesday 
9:00–10:00  Roger HeathBrown 
Counting rational points on lines and conics
Suppose one wants to count points (x,y) on a line ax+by=c, in which x=u/w and y=v/w are rational numbers. One might require that u,v,w are integers with no common factor, and count solutions with u,v,w ≤ B. How does this number grow as B tends to infinity? This question is fairly easy and “classical”. It turns out that the counting function is roughly C_{a,b,c}B, for some positive constant C_{a,b,c}. However to see this behaviour, B must be large enough compared to the coefficients a,b,c; and it turns out that the point at which the correct asymptotic behaviour begins will depend on the size of the smallest solution to the original equation ax+by=c. All this will be described in the talk, which will go on to discuss the analogous situation for conics.

10:00–10:30  Coffee and discussion 
10:30–11:30  Daniel Loughran 
Rational points on fibrations I
In this talk I describe some progress towards Serre’s problem on the number of
varieties in a family with a rational point using tools from additive combinatorics. This
is joint work with Lilian Matthiesen.

11:30–12:30  Lilian Matthiesen 
Rational points on fibrations II
In this talk I will first discuss a general asymptotic results on linear
correlations of multiplicative functions that satisfy certain natural conditions. A simple
example of the type of correlations that we consider would be
Σ_{n,d<x} b(n) b(n+d) … b(n+rd),
where b denotes the characteristic function of sums of two squares.
In the second part, we then discuss how this result is used within the joint project with
Dan Loughran on the number of varieties in a family with a rational point.

12:30–14:30  Lunch and discussion 
14:30–15:30  Trevor Wooley 
Counting generic rational points on varieties
We consider the number of rational points on a variety with appropriate height
function in a large box – specifically emphasising the generic points. We obtain upper
bounds in the case of systems of diagonal equations via recent work on relatives of
Vinogradov’s mean value theorem (via efficient congruencing and decoupling). There are connections with certain biprojective varieties.

15:30–16:15  Coffee and discussion 
16:15–16:45  Lasse Grimmelt 
Vinogradov’s Theorem with Fouvry Iwaniec primes
Fouvry and Iwaniec showed that there are infinitely many primes that are a sum of a square
and a prime square. In this talk a proof of Vinogradov’s three prime theorem restricted
to these primes is outlined. The main techniques used are a transference principle version
of the circle method, the linear sieve with sieve switching, and the combinatorical
dissection of sums over primes given in a work of Duke, Friedlander, and Iwaniec.

17:00–17:30  Erik VisseMartindale 
Serre’s problem for conic bundles over low degree hypersurfaces
In recent years there has been a lively interest in Serre’s problem on the density of fibres that have a rational point among a family of varieties. The literature contains a wide range of results varying from upper and lower bounds to asymptotics, and from simpler to more complicated bases over which the families are defined.
In joint work with Efthymios Sofos, we analyse some conic bundles over general hypersurfaces of low degree. Using circle method results of Birch in combination with sieving methods, we manage to find asymptotics for the number of fibres of bounded height containing a rational point. Moreover, we are able to factor the leading constant in padic parts, obtaining similar behaviour as observed by Loughran for other families of varieties.

17:30–  Dinner and free time 
Wednesday 
9:00–10:00  Jennifer Park 
Cycles in supersingular lisogeny graphs and corresponding endomorphisms
The problem of computing endomorphism rings of supersingular elliptic curves is an important one in postquantum cryptography. In this talk, we expand on the work of Kohel and study the cycles in supersingular lisogeny graphs that could generate the endomorphism rings (or a fullrank suborder) of supersingular elliptic curves. This work is joint with Efrat Bank, Catalina CamachoNavarro, Kirsten Eisentraeger, and Travis Morrison.

10:00–10:30  Coffee and discussion 
10:30–11:30  Jordan Ellenberg 
Counting points, counting fields, and heights on stacks I
A folklore conjecture is that the number N_{d}(K,X) of degreed
extensions of K with discriminant at most d is on order c_{d} X. In the case K =
Q, this is
easy for d=2, a theorem of Davenport and Heilbronn for d=3, a much harder theorem of
Bhargava for d=4 and 5, and completely out of reach for d>5. More generally, one can ask
about extensions with a specified Galois group G; in this case, a conjecture of Malle
predicts that the asymptotic growth is on order X^{a}(log X)^{b} for specified constants a,b.
The form of Malle’s conjecture is reminiscent of the Batyrev–Manin conjecture, which says
that the number of rational points of height at most X on a Batyrev–Manin variety also
grows like X^{a}(log X)^{b} for specified constants a,b. What’s more, an extension of
Q with
Galois group G is a rational point on a Deligne–Mumford stack called BG, the classifying
stack of G. A natural reaction is to say “the two conjectures is the same; to count number
fields is just to count points on the stack BG with bounded height?” The problem: there is
no definition of the height of a rational point on a stack. I’ll explain what we think the
right definition is, and explain how it suggests a heuristic which has both the Malle
conjecture and the Batyrev–Manin conjecture as special cases.

11:30–12:30  David ZureickBrown 
Counting points, counting fields, and heights on stacks II
A folklore conjecture is that the number N_{d}(K,X) of degreed
extensions of K with discriminant at most d is on order c_{d} X. In the case K =
Q, this is
easy for d=2, a theorem of Davenport and Heilbronn for d=3, a much harder theorem of
Bhargava for d=4 and 5, and completely out of reach for d>5. More generally, one can ask
about extensions with a specified Galois group G; in this case, a conjecture of Malle
predicts that the asymptotic growth is on order X^{a}(log X)^{b} for specified constants a,b.
The form of Malle’s conjecture is reminiscent of the Batyrev–Manin conjecture, which says
that the number of rational points of height at most X on a Batyrev–Manin variety also
grows like X^{a}(log X)^{b} for specified constants a,b. What’s more, an extension of
Q with
Galois group G is a rational point on a Deligne–Mumford stack called BG, the classifying
stack of G. A natural reaction is to say “the two conjectures is the same; to count number
fields is just to count points on the stack BG with bounded height?” The problem: there is
no definition of the height of a rational point on a stack. I’ll explain what we think the
right definition is, and explain how it suggests a heuristic which has both the Malle
conjecture and the Batyrev–Manin conjecture as special cases.

12:30–  Lunch and free afternoon 
Thursday 
9:00–10:00  Ulrich Derenthal 
Ominimality and Cox rings over number fields for Manin’s conjecture
Manin’s conjecture predicts the asymptotic behavior of the number of rational points of bounded height on Fano varieties over number fields. We prove this conjecture for a family of nonsplit singular quartic del Pezzo surfaces over arbitrary number fields. For the proof, we parameterize the rational points on such a del Pezzo surface by integral points on a nonuniversal torsor (which is determined explicitly using a Cox ring of a certain type), and we count them using a result of Barroero–Widmer on lattice points in ominimal structures. This is joint work in progress with Marta Pieropan.

10:00–10:30  Coffee and discussion 
10:30–11:00  Diego Izquierdo 
Vanishing theorems and Brauer–Hasse–Noether exact sequence for higherdimensional fields
When one wants to study the arithmetic of a given field K, it is often useful to understand the cohomology of the Galois module of roots of unity Q/Z(1) or, more generally, the cohomology of its twists Q/Z(r). In this talk, we will be interested in the situation when K is a finite extension of the Laurent series field in m variables k((x_{1},…,x_{m})) with coefficients in a finite field, a padic field or a number field. We will discuss some vanishing theorems as well as some exact sequences that play the role of the Brauer–Hasse–Noether exact sequence for the field K.

11:15–12:15  Marta Pieropan 
Rational points over C_{1} fields of characteristic 0
In the 1950s Lang studied the properties of C_{1} fields, that is, fields over
which every hypersurface of degree at most n in a projective space of dimension n has a
rational point. Later he conjectured that every smooth proper rationally connected variety
over a C_{1} field has a rational point. I will explain how to find rational points on some
Fano threefolds over C_{1} fields of characteristic 0.

12:15–14:30  Lunch and discussion 
14:30–15:30  Sho Tanimoto 
Geometric consistency of Manin’s Conjecture
Manin’s Conjecture predicts an asymptotic formula for the counting function of
rational points on a Fano variety after removing an exceptional set. We propose the
construction of the exceptional set and prove that it is contained in a thin subset of
rational points using the boundedness of singular Fano varieties and Hilbert
Irreducibility Theorem. This is joint work with Brian Lehmann and Akash Sengupta.

15:30–16:15  Coffee and discussion 
16:15–16:45  Jackson Morrow 
Irrational points on random hyperelliptic curves
Let d and g be positive integers with 1<d<g. If d is odd, we prove
there exists B(d)>0 such that a positive proportion of odd genus g hyperelliptic
curves over Q have at most B(d) points of degree d. If d is even, we
similarly bound the degree d points not pulled back from degree d/2 points of the
projective line. Our proof proceeds by refining Park’s recent application of tropical
geometry to symmetric power Chabauty, and then applying results of Bhargava and Gross on
average ranks of Jacobians of hyperelliptic curves. This is joint work with Joseph
Gunther.

17:00–17:30  Marc Paul Noordman 
Lower bounds for gonality using bad reduction
When studying the possible torsion structures of elliptic curves over number
fields of a fixed degree, one is naturally led to the question of existence of points of
low degree on modular curves. This question is related to the gonality of such curves,
defined as the lowest degree of a rational map from the curve to the projective line, by a
theorem of Frey: if a smooth projective curve over Q has infinitely many points of degree
d, then there exists a nonconstant map from this curve to the projective line of degree
at most 2d. Hence, lower bounds on the gonality of a curve imply results on the finiteness
of points of low degree on that curve.
However, in general it is a hard problem to compute the gonality of a given curve. In this
talk I will outline a strategy to obtain lower bounds on the gonality of a modular curve,
by using reduction modulo primes dividing the level of the curve. This reduction is not
smooth, but often admits a detailed geometric description obtained by studing ptorsion on
elliptic curves in characteristic p. This allows us, amongst others, to prove that there
are only finitely many elliptic curves defined over number fields of degree 7 whose
torsion structure is isomorphic to Z/2Z × Z/2Z, partially answering a question by Derickx
and Sutherland.
This is joint work with Maarten Derickx and Bas Edixhoven.

17:30–17:45  Conference photo (on the hill behind the Dorpshuis) 
17:45–  Dinner and free time 
Friday 
9:00–10:00  Brendan Creutz 
There are no transcendental Brauer–Manin obstructions on abelian varieties
Elements in the Brauer group of a variety over a number field which are annihilated by base change to an algebraic closure are said to be algebraic. In general it is possible for the Brauer group to provide an obstruction to the Hasse Principle or Weak Approximation which cannot be explained by algebraic classes. I will describe a recent result that such transcendental obstructions do not occur for torsors under abelian varieties.

10:00–10:30  Coffee and discussion 
10:30–11:30  Alexei Skorobogatov 
Brauer groups of diagonal quartic surfaces over Q
Algebraic Brauer groups of diagonal quartic surfaces have been classified by Martin
Bright. To determine their transcendental Brauer groups over Q one first needs to
understand the Galois representation on the transcendental part of the second cohomology
group. This can be done more generally for K3 surfaces with complex multiplication. A more
subtle problem is to determine the image of the Brauer group in the geometric Brauer
group. This can be done using some homological algebra from an earlier joint work of
JeanLouis ColliotThélène and the speaker. The case of odd order torsion was settled by
E. Ieronymou and the speaker. It turns out that the 2primary subgroup of the Brauer group
of any diagonal quartic surface over Q is algebraic, with the exception of only two cases
that can be explicitly described. This is a joint work with Damián Gvirtz.

11:30–  Lunch and free afternoon 