Rational Points - Geometric, Analytic and Explicit Approaches

Schedule

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Martin Bright (American University of Beirut)
Bad reduction of the Brauer-Manin obstruction

It is increasingly recognised that the Brauer group and the associated Manin obstruction on a variety over a number field are intimately related to the geometry of the variety at places of bad reduction. I will describe recent work showing how the geometry influences both the Brauer group and the obstruction, with particular regard to del Pezzo surfaces.

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Tim Browning (Bristol)
Rational points on the intersection of a cubic and quadric hypersurface

We investigate the smooth Hasse principle for complete intersections cut out by a cubic and quadric hypersurface, both defined over the rationals. This is joint work with Rainer Dietmann and Roger Heath-Brown.

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Nils Bruin (Simon Fraser University)
TBA:Torsion in ideal class groups By Algebraic geometry

When people do explicit computations with Mordell-Weil groups, one ingredient is usually the computation of unit and class groups of number fields. We will consider the opposite direction: how Abelian varieties with special properties give rise to number fields with a prescribed part of the class group. These ideas have been used by various authors in various degrees of generalities. In this preliminary report I will describe some of those ideas and some of the relations between them. This is joint with Victor Flynn and Damiano Testa.

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Jean-Louis Colliot-Thélène (CNRS et Université Paris-Sud)
Rational points on one-parameter families of 2-dimensional quadrics

On these much studied varieties, we shall compare the result of application of various techniques, in particular descent and the formal lemma.

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Ulrich Derenthal (München)
Integral Brauer-Manin obstructions for sums of two squares and a power

Legendre and Gauß characterized integers that are sums of three squares. Vaughan conjectured that an integral Hasse principle holds for the representability of sufficiently large integers as sums of two squares and a $k$-th power. For $k=9$, Jagy and Kaplanski gave an elementary counterexample to this conjecture: $x^2+y^2+z^9=m$ has no integral solution $(x,y,z)$ for certain m even though local solutions exist. For arbitrary $k>2$, Dietmann and Elsholtz gave similar counterexamples that can be interpreted as failures of strong approximation. In this talk, I will discuss recent work of Fabian Gundlach (LMU München) that explains and generalizes this systematically using integral Brauer-Manin obstructions.

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Tim Dokchitser (Bristol)
L-functions of hyperelliptic curves

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Tom Fisher (Cambridge)
Computing the Cassels-Tate pairing

I will describe joint work with Rachel Newton on computing the Cassels-Tate pairing on the 3-Selmer group of an elliptic curve. This generalises the method for computing the pairing on the 2-Selmer group in Cassels' paper "Second descents for elliptic curves". Cassels' method involves (i) identifying a certain local pairing with the Hilbert norm residue symbol and (ii) finding rational points on conics defined over number fields. Both parts require significant modification in our case, and the analogue of (ii) is currently only practical in small examples.

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David Harari (Orsay)
Linear algebraic groups over the function field of a p-adic curve

Let $K$ be the function field of a $p$-adic curve, e.g. $K=\mathbf{Q}_p(t)$. Let $G$ be a linear algebraic group over $K$. We discuss local global properties for $G$, using Galois cohomology (joint work with T. Szamuely).

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David Holmes (Leiden)
Rational points on Kummer varieties

Let $A$ be an abelian variety over a number field, and $K=A/\pm1$ the associated Kummer variety. If the dimension of $A$ is 2, $K$ is an example of a K3 surface. The rational points on $K$ are closely related to the rational points on $A$ over quadratic extensions of the base field. We use this to gain information on the distribution of rational points on $K$. See arXiv:1211.4597.
This is joint work with René Pannekoek.

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David M^cKinnon (Waterloo)
Theorems of Roth and Liouville, only bigger

How well can rational points approximate algebraic points on an algebraic variety? The answer, obviously, is “it depends”. There has been much work done in connecting this question with the curvature of the variety in question, but mostly the results have been global, relating global distribution of points with global measures of curvature. In my talk, I will discuss some local relationships, between approximations constants and Seshadri constants, which are a geometric measure of local positivity.

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Jan Steffen Müller (Hamburg)
$p$-adic heights and integral points on hyperelliptic curves

Let $X$ be a hyperelliptic curve of genus $g$ over the rationals. I will describe a method to $p$-adically approximate integral points on $X$ when the Mordell-Weil rank of the Jacobian $J$ of $X$ is equal to $g$. For this method we express certain $p$-adic heights on $J$ in terms of iterated Coleman integrals and intersection multiplicities. This is joint work with Jennifer Balakrishnan and Amnon Besser.

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Jonathan Pila (Oxford)
On atypical intersections in powers of modular curves

The Zilber-Pink conjecture concerns suitable ambient varieties $X$ equipped with a collection of “special subvarieties”. For a subvariety $V$ of $X$ the conjecture governs those intersections of $V$ with special subvarieties which are atypical in dimension. It implies well-known theorems/conjectures such as Raynaud's theorem (aka the Manin-Mumford conjecture) and the Andre-Oort conjecture but goes far beyond them. I will describe some partial results and some conditional results in this direction, all joint work with Philipp Habegger.

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Jeroen Sijsling (Warwick)
Galois descent and rational points

Let $L/k$ be an extension of fields. Let $C$ be an algebraic curve over $L$, defined as the zero locus of a set of polynomials $\{ f_1, \ldots , f_n \}$. Suppose that for all automorphisms $\sigma$ of $L$ fixing $k$, the conjugate curve $C^\sigma$ (which is simply the zero locus of the conjugated set of polynomials $\{ f_1^{\sigma} , \ldots , f_n^{\sigma} \}$) is isomorphic over $L$ with $C$. Does there then exist a curve $C_0$ over $k$ that becomes isomorphic with $C$ over $L$?

Contrary to what one may intuitively expect, the answer to this question is usually not affirmative. The obstruction is a cohomological one, but it is a challenge to make it explicit and amenable to computation. We will give a complete answer for hyperelliptic curves and plane quartics, and we will see how this answer relates to the presence of rational points on certain varieties associated to the curve. This is joint work with Reynald Lercier and Christophe Ritzenthaler.

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Alexei Skorobogatov (Imperial)
Diagonal quartic surfaces, complex multiplication, and the Brauer-Manin obstruction

This is a joint work with Evis Ieronymou. We use isogenies of K3 and abelian surfaces to reduce the calculation of the odd order subgroup of the Brauer group of diagonal quartic surfaces over $\mathbf{Q}$ to certain Galois representations attached to elliptic curves with complex multiplication by the ring of Gaussian numbers. Our methods also allow to calculate the corresponding Brauer-Manin obstruction.

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Maciej Ulas (Jagellonian)
Rational solutions of certain Diophantine equations involving norms

In this talk we will present some results concerning the unirationality of the algebraic variety $S_f$ given by the equation

\[S_{f}: N_{K/k} (X_{1},X_{2},X_{3}) = f(t) , \]

where $K/k$ is (mainly) a pure cubic extension and $f \in k[t]$. We prove that if $\deg f = 4$ and the variety $S_f$ contains a $k$-rational point $(x_{0},y_{0},z_{0},t_{0})$ with $f(t_{0}) \neq 0$, then $S_{f}$ is $k$-unirational. Similar result will be proved for a broad family of quintic polynomials $f$ satisfying some mild conditions (for example this family contains all irreducible polynomials). Moreover, unirationality of $S_f$ will be proved for any monic polynomial $f$ of degree 6 satisfying the condition $f(t) \neq f(\zeta t)$, where $\zeta$ is primitive third root of unity. The last result give a partial answer on a recent question posed by Várilly-Alvarado and Viray.

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Ronald van Luijk (Leiden)
Density of rational points on Del Pezzo surfaces of degree one

We state conditions under which the set of rational points on a Del Pezzo surface of degree one over an infinite field is Zariski dense. For example, it suffices to require that the elliptic fibration induced by the anticanonical map has a nodal fiber over a rational point of the projective line. It also suffices to require the existence of a rational point that does not lie on six exceptional curves of the surface and that has order three on its fiber of the elliptic fibration. This allows us to show that within a parameter space for Del Pezzo surfaces of degree one over the real numbers, the set of those surfaces defined over the rational numbers for which the set of rational points is Zariski dense, is dense with respect to the real analytic topology. We also state conditions that may be satisfied for every del Pezzo surface and that can be verified with a finite computation for any del Pezzo surface that does satisfy them. This is joint work with Cecília Salgado.

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Bianca Viray (Brown)
Two-torsion Brauer classes on double covers of ruled surfaces

Let $X$ be a smooth double cover of a geometrically ruled surface over a field of characteristic different from 2. We give a finite presentation of the two-torsion of the transcendental Brauer group of $X$ with generators given by central simple algebras over the function field of $X$ and relations coming from the Néron-Severi group of $X$. This enables one to compute the transcendental Brauer group as a Galois module and, in some cases, to compute the image of ${\rm Br} \, X$ in ${\rm Br} \, \overline{X}$. This is joint work with Brendan Creutz.

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Olivier Wittenberg (CNRS/ENS)
On a conjecture of Kato and Kuzumaki about $p$-adic fields

The field of $p$-adic numbers is not $C_2$. In 1986, Kato and Kuzumaki introduced two variants of the $C_2$ property, denoted $C_2^0$ and $C_1^1$, conjectured that they hold in the case of $p$-adic fields, and gave a proof for hypersurfaces of prime degree. In this talk we will prove that $p$-adic fields are $C_1^1$. Equivalently, if $K$ is a $p$-adic field and $X$ is a Fano hypersurface over $K$, any element of $K$ may be written as a product of norms from the residue fields of the closed points of $X$.

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Trevor Wooley (Bristol)
Systems of diagonal cubic equations

We describe recent work, joint work with Joerg Bruedern, which delivers the Hasse Principle for systems of $r$ diagonal cubic equations in general position having at least $6r+1$ variables. This conclusion attains the sharpest constraint on the number of variables permitted by the (square-root) convexity barrier in the Hardy-Littlewood method. In certain circumstances, one may obtain conclusions beating this convexity barrier, and if time permits we will describe some such result.