Schedule for: 18w5012 - Rational and Integral Points via Analytic and Geometric Methods

We consider a family of abelian varieties over a number field $K$ , i.e. a variety $X$ with a map to a curve $B$ whose fibres are abelian varieties (the interesting cases are when $B$ is the projective line or an elliptic curve with positive rank). The generic fibre is an abelian variety over the function field $K(B)$ and the group of $K(B)$-rational points has a rank $r$. For almost all points $t$ in $B(K)$ the fibre is an abelian variety $X_t$ over $K$ and the group of $K$-rational point has rank $r(t)$. A specialisation theorem of Silverman says that for or almost all points $t$ in $B(K)$ the rank $r(t)$ is greater or equal to $r$. We want to understand the distribution of $r(t)$, in particular we ask wether there are infinitely many $t$'s 1) with $r(t)=r$, 2) with $r(t)>r$. The problem looks very hard in general, but, under specific geometric conditions, we will settle the second question, and provide interesting example where much more can be proven. This is a joint work with Cecília Salgado.

The classical Schinzel's hypothesis and its quantitative version, the Bateman-Horn's conjecture, states that a system of polynomials in one variable takes infinitely many simultaneously prime values under some necessary assumptions. We will present in this talk a proof of a generalization of these conjectures to the case of a integer form in many variables. In particular, we will establish that a polynomial in moderately many variables takes infinitely many prime (but also squarefree) values under some necessary assumptions. The proof relies on the Birch's circle method but can be achieved in 50% fewer variables than in the classical Birch setting. Moreover this result can be applied to study the Hasse principle and weak approximation for some normic equations.

We show that the uniform boundedness of the transcendental Brauer group of K3 surfaces and abelian varieties of bounded dimension defined over number fields of bounded degree is a consequence of a conjecture of Coleman about rings of endomorphisms of abelian varieties. We also show that this conjecture of Coleman implies the conjecture of Shafarevich about the N\'eron-Severi lattices of K3 surfaces. This is a joint work with Martin Orr and Yuri Zarhin.

What can we say about the variation of the rank in a family of elliptic curves ? We know in particular that if infinitely many curves in the family have non-zero rank, then the set of rational points is Zariski dense in the associated elliptic surface. We use a “conjectural substitute” for the geometric rank (or rather for its parity) : the root number. For a non-isotrivial family, under two analytic number theory conjectures I show that the root number is -1 (resp. +1) for infinitely many curves in the family. On isotrivial families however, the root number may be constant : I describe its behaviour in this case.

Weak and strong approximation for algebraic groups are established by using arithmetic duality. In my talk, I will talk about how to apply arithmetic duality to weak and strong approximation for some special group schemes over the projective line.

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 $g$ hyper elliptic curves over $\mathbf{Q}$ 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.

A topic of current interest regards 'how often' a variety has a rational point. This topic was initially studied by Serre who gave upper bounds in the case of families of conics parametrised by a projective space. In the last few years this topic has been significantly enriched by Loughran and others. I will begin by giving an overview of the latest developments and finish by discussing joint work with Erik Visse, where asymptotics are given for a family of conics parametrised by arbitrary smooth hypersurfaces of low degree.

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 rationally connected threefolds over $C_1$ fields of characteristic 0.

A famous theorem due to Erdős and Kac states that the number of prime divisors of an integer N behaves like a normal distribution. In this talk we consider analogues of this result in the setting of arithmetic geometry, and obtain probability distributions for questions related to local solubility of algebraic varieties. This is joint work with Efthymios Sofos.

The notion of Campana points has been introduced by Abramovich and Varilly-Alvarado and this interpolates between rational points and integral points. We study a variant of this notion and prove Manin’s conjecture for Campana points on equivariant compactifications of vector groups using the height zeta functions method. As an outcome we propose log Manin’s conjecture for klt Campana points. This is joint work with Tony Varilly-Alvarado.

Given a curve of genus at least 2, it was proven in 1983 by Faltings that it has only finitely many rational points. Unfortunately, this result is ineffective, in that it gives no bound on the number of rational points. 40 years earlier, Chabauty proved the same result under the condition that the rank of the Jacobian of the curve is strictly smaller than the genus. While this is obviously a weaker result, the methods behind that proof could be made effective, and this was done by Coleman in 1985. Coleman's work led to a procedure known as the Chabauty-Coleman method, which has shown to be extremely effective at determining the set of rational points exactly, particularly in the case of hyperelliptic curves. In this talk I will discuss how we implement this method using Magma and Sage to provably determine the set of rational points on a large set of genus 3, rank 1 hyperelliptic curves, and how these calculations fit into the context of the state of the art conjectures in the field. The subject of this talk is joint work with Jennifer Balakrishnan, Francesca Bianchi, Victoria Cantoral-Farfan, and Mirela Ciperiani.

We propose a 2-cover method to study rational points on elliptic surfaces. We apply it to several isotrivial Kummer-type families whose generic Mordell-Weil ranks are 0 so that geometric argument may fail and we show that for these families rational points are Zariski dense and even dense in real topology, which is thereby in favor of a conjecture of Mazur.

A classical result of Colliot-Thélène and Sansuc states that the only obstruction to the Hasse principle and weak approximation for generalised Châtelet surfaces is the Brauer-Manin one, conditionally on Schinzel's hypothesis. Inspired by their work, we study the analogous questions concerning the existence and the density of integral points on the corresponding affine surfaces, again under Schinzel's hypothesis. To be precise, we show that the Brauer-Manin obstruction is the only obstruction to the integral Hasse principle for an infinite family of generalised affine Châtelet surfaces. Moreover, we show that the set of integral points on any surface in this family satisfies a strong approximation property off infinity with Brauer-Manin obstruction.

Let $S$ be a del Pezzo surface of degree $1$ over a number field $k$. The main goal of this talk is to give easily verifiable sufficient conditions under which its set $S(k)$ of rational points is Zariski dense. It is well known that almost all fibers of the anticanonical map $\varphi \colon S \dashrightarrow \mathbb{P}^1$ are elliptic curves with the unique base point of $\varphi$ as zero. Suppose that $P \in S(k)$ is a point of finite order $n>1$ on its fiber. Then there is another elliptic fibration on the blow-up of $S$ at $P$. We will see where it comes from and how it can be used to define a proper closed subset $Z \subset S$ such that (1) it is easy to verify for any point on $S$ whether it lies in $Z$, and (2) the set $S(k)$ contains a point outside $Z$ if and only if $S(k)$ is Zariski dense. In other words, if $S(k)$ is Zariski dense, then we can prove this by exhibiting a rational point outside $Z$. We will also compare this to previous work. This is joint work with Jelle Bulthuis inspired by an example of Noam Elkies.

Let l be a positive integer. We discuss improved average bounds for the l-torsion of the class groups for some families of number fields, including degree-d-fields for d between 2 and 5. The improvements are based on refinements of a technique due to Ellenberg, Pierce and Wood. This is joint work with Martin Widmer.

Masser and Vaaler gave an asymptotic formula for the number of algebraic numbers of given degree and increasing height. This problem was solved by counting lattice points in an expanding star body. We'll explain how to estimate the volumes of slices of star bodies, which allows one to count algebraic integers, algebraic integers of given norm and/or trace, and more. There will be pictures. This is joint work with Robert Grizzard.

After fixing numerical invariants such as dimension, it is natural to ask which birational classes of varieties fail the Hasse principle, and moreover whether the Brauer group (or certain distinguished subsets) explains this failure. In this talk, we will focus on K3 surfaces (e.g. a double cover of the plane branched along a smooth sextic curve), which have been a testing ground for many conjectures on rational points. In 2014, Ieronymou and Skorobogatov asked whether any odd torsion in the Brauer group of a K3 surface could obstruct the Hasse principle. We answer this question in the affirmative for transcendental classes; via a purely geometric approach, we construct a 3-torsion transcendental Brauer class on a degree 2 K3 surface over the rationals with geometric Picard rank 1 (hence with trivial algebraic Brauer group) which obstructs the Hasse principle. Moreover, we do this without needing a central simple algebra representative. This is joint work with Tony Varilly-Alvarado.

It is widely expected that, if a curve over a global field has no rational points, that there is an obstruction to existence of rational points coming from the Brauer group. One piece of evidence for this is an heuristic due to Poonen. We show that Poonen's argument also applies to p-primary subgroups of the Brauer group (for any prime p) but that there are examples of curves with no rational point but not having an obstruction coming from the p-primary subgroups of the Brauer group. Joint work with B. Creutz and B. Viray.

This is joint work with Tim Browning. The talk concerns the variety $X_1Y_1^2+X_2Y_2^2+X_3Y_3^2+X_4Y_4^2=0$ in $P^3 \times P^3$. The height of a point $(x,y)$ is given by $H(x)^3H(y)^2$, and Manin's conjecture predicts asymptotically $cB \log B$ points of height at most $B$. To obtain this one must exclude points on the subvariety $x_1x_2x_3x_4=0$; but in order to achieve the Peyre constant one must exclude an infinite number of subvarieties in which $x_1x_2x_3x_4$ is a square. By combining the circle method with lattice point counting techniques we are able to prove Manin's conjecture for this example, and the talk will give an overview of the various ingredients, and the way that they fit together.

We discuss small solutions to ternary diagonal inequalities of any degree where all of the variables are assumed to be of size P. We study this problem on average over a one-parameter family of forms and discuss a generalization of work of Bourgain on generic ternary diagonal quadratic forms to higher degree. In particular we discuss how these Diophantine inequalities are related to counting rational points close to varieties.

There has been much interest recently in bounding the Brauer groups of K3 surfaces over number fields. On the other hand, the arithmetic of integral points on log K3 surfaces appears to share some features with that of rational points on K3 surfaces. Some of the simplest examples of log K3 surfaces are the open surfaces obtained by starting with a projective del Pezzo surface and removing a smooth anticanonical divisor. We use Merel's boundedness of torsion on elliptic curves to prove boundedness of the Brauer groups of such log K3 surfaces over a number field. This is joint work with Julian Lyczak.

A folklore conjecture is that the number $N_d(K,X)$ of degree-$d$ extensions of $K$ with discriminant at most $d$ is on order $c_d X$. In the case $K = \mathbb{Q}$, 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 holds 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 $\mathbb{Q}$ 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. This is joint with Jordan Ellenberg and Matt Satriano.

We will prove that smooth Quartic hypersurfaces satisfy the Hasse Principle as long as they are defined over at least 30 variables. The key tool here is employing Kloosterman's version of circle method. This is a joint work with Oscar Marmon (U Lund).

Yongqi Liang has shown that for rationally connected varieties over a number field K, sufficiency of the Brauer-Manin obstruction to the existence of rational points over all finite extensions of K implies sufficiency of the Brauer-Manin obstruction to the existence of zero-cycles of degree 1 over K. I will discuss joint work with Francesca Balestrieri where we extend Liang's result to Kummer varieties.

For a nice algebraic variety $X$ over a number field $F$, one of the central problems of Diophantine Geometry is to locate precisely the set $X(F)$ inside $X(\mathbb{A}_F)$, where $\mathbb{A}_F$ denotes the ring of adèles of $F$. One approach to this problem is provided by the finite descent obstruction, which is defined to be the set of adelic points which can be lifted to twists of torsors for finite étale group schemes over $F$ on $X$. More recently, Kim proposed an iterative construction of another subset of $X(\mathbb{A}_F)$ which contains the set of rational points. We compare the two constructions. Our main result shows that the two approaches are equivalent.

We prove that every Brauer class over a field splits over a torsor under an abelian variety. If the index of the class is not congruent to 2 modulo 4, we show that the Albanese variety of any smooth curve of positive genus that splits the class also splits the class. This can fail when the index is congruent to 2 modulo 4, but adding a single genus 1 factor to the Albanese suffices to split the class. This is joint work with Max Lieblich.

For an abelian variety A over a number field K, a consequence of the Birch and Swinnerton-Dyer conjecture is the 2-parity conjecture: the global root number agrees with the parity of the 2-infinity Selmer rank. It is a standard result that the root number may be expressed as a product of local terms and we show that, over any quadratic extension of K, the same holds true for the parity of the 2-infinity Selmer rank. Using this we prove several new instances of the 2-parity conjecture for general principally polarised abelian varieties by comparing the local contributions arising. Somewhat surprisingly, the local comparison relies heavily on results from the theory of quadratic forms in characteristic 2.

Already 190 years ago Jacobi in a paper in Crelle's journal described the celebrated closure theorem of Poncelet as a ``bekanntes Problem der Elementargeometrie''. Some natural number theoretical questions arising in this context, will be discussed during this talk.

(Joint work with Yonatan Harpaz.) The Brauer-Manin obstruction is expected to control the existence and weak approximation properties of rational points on homogeneous spaces of linear algebraic groups over number fields. We establish the zero-cycle variant of this conjecture. The same method also leads to a new proof of Shafarevich's theorem that finite nilpotent groups are Galois groups over any number field.