Alan Baker: The abc-conjecture: a brief survey
The abc-conjecture is a simple statement about integers that encapsulates many important results and has come to be seen as one of the key problems for the future direction of mathematics. In this talk we shall survey our current state of knowledge in the field.
Massimo Bertolini: Rational points on elliptic curves and values of complex and p-adic L-functions
The Birch and Swinnerton-Dyer (BSD) conjecture relates the group of rational points on an elliptic curve to the complex L-function of the curve. After recalling basic facts and results in the theory, I plan to describe some recent developments on the BSD conjecture, which show that non-archimedean (as well as archimedean) techniques play a role in the study of this problem.
Tim Browning: Cubic divisor problems and applications to the Manin conjecture
A classical topic in analytic number theory concerns the average order of the divisor function as it ranges over the values of polynomials. In this talk I will discuss a correspondence between this problem for reducible binary forms of degree 3 and the distribution of rational points on Fano varieties. Specifically I will give a new proof of the Manin conjecture for certain bilinear hypersurfaces, and furthermore, for the split non-singular del Pezzo surface of degree 5. The latter result is joint work with Per Salberger.
Jörg Brüdern: Amalgamist's delight. Some cases of the Manin-Peyre Conjecture(s)
This talk surveys recent work circumscribed by the title line, mostly joint with Valentin Blomer. Various families of projective varieties are considered, and the Manin-Peyre conjectures are demonstrated for these by a blend of analytic and more elementary techniques that are perhaps not all new individually, but the mix is. Particular attention will be paid to sums-of-fractions varieties. In all cases where our method worked (so far), an analytic continuation for the height zeta function came along for free. If time permits, an attempt will be made to delight the amalgamist again while contrasting the results with some prime paucity estimates also obtained in collaboration with Blomer.
Nils Bruin: 2-descent on Jacobians of smooth plane quartics
Relative to hyperelliptic curves, the computational accessibility of arithmetic properties of more general curves and their jacobians leaves much to be desired. I will report on the joint efforts with Bjorn Poonen and Michael Stoll to improve this situation.
Tim Dokchitser: Parity predictions for ranks of elliptic curves
I will review the status of the Birch-Swinnerton-Dyer and Parity Conjecture for elliptic curves, and discuss some examples and consequences of the standard `minimalistic conjecture' for ranks of elliptic curves in families.
Jordan Ellenberg: Analytic number theory and spaces of rational curves
(joint work with A. Venkatesh) Suppose X is a smooth Fano variety, and K a global field. Conjectures of Batyrev-Manin and Peyre give very precise predictions for the asymptotic behavior of the function NX/K(B) which enumerates points of X(K) of height at most B. When K is Fq(t), these conjectures are naturally related to questions about the geometry of the space of rational curves on X. I will try to promote a general philosophy that nice asymptotic behavior of the type predicted by Batyrev-Manin should correspond with stabilization of cohomology for spaces of rational curves. In a particularly favorable situation, where X is a very low-degree Fermat hypersurface, the Batyrev-Manin conjecture can be verified by traditional methods of analytic number theory, and we explain how to use this fact to prove irreducibility of the space of rational curves on X.
Andreas-Stephan Elsenhans: Arithmetic of cubic surfaces
We will take a look on the arithmitic properties of cubic surfaces. The main focus will be on 27 lines and the Galois group action on them. Different descriptions of the moduli space of cubic surfaces are used to construct several Galois groups. Finally we will inspect the Manin conjecture for these surfaces.
Clemens Fuchs: Rational points parametrizing composite lacunary rational functions
Let f be a function that is the composite of two rational functions g, h. In this talk we discuss the following question: What can be said about the composition factors g, h when we assume that the number of distinct zeros and poles of f is fixed? We give a characterization of all such functions and decompositions in terms of the rational points of finitely many effectively computable algebraic varieties. The result can be seen as a multiplicative analogue of a theorem due to Zannier on polynomials which have a fixed number of non-constant terms. This is joint work with Attila Pethö.
Nicholas Katz: Sato-Tate theorems for finite-field Mellin transforms
Kiran Kedlaya: Numerical p-adic integration and (potential) applications to rational points
Coleman introduced an integration theory for 1-forms on p-adic analytic curves, in which Frobenius actions are used to perform analytic continuation. This theory can often be made computationally effective; I'll indicate how this is done for hyperelliptic curves, and illustrate (with some toy examples) how this can be used to find rational points using either the standard Chabauty method or some nonabelian variants. Joint work with Jennifer Balakrishnan and Minhyong Kim.
Ronald van Luijk: Computing Picard groups
In general it is hard to find the Picard group of a given surface defined over a number field. I will start by describing a method that computes at least the rank in many cases. We will also describe various improvements by Elsenhans and Jahnel that have been made to this method and conclude with the latest related trick, by Shioda and Schütt, which allows us to find not only the rank, but also generators for the Picard group.
Emmanuel Peyre: Counting with versal torsors
As was predicted by Per Salberger, versal torsors are very efficient tools to prove the principle of Batyrev and Manin for points of bounded height on Fano varieties. The aim of this talk is to explain this technique with several simple examples.
Per Salberger: Density of rational points on covers of Pn
We report on recent progress on a conjecture of Serre concerning the density of rational points on covers of Pn.
N. Saradha: A problem on residue systems
Given an integer k, can the first phi(k) primes, coprime to k form a reduced residue system? In this talk, we investigate a conjecture of Pomerance on this question. We show how theory and experiment go hand in hand in pursuit of an answer. (Joint work with Lajos Hajdu)
Andrew Sutherland: L-polynomial distributions of genus 2 curves
Given a smooth projective curve C/Q, we may consider the distribution of the Lp(p-1/2T) as p varies over primes of good reduction, where the polynomial Lp(T) the numerator of the zeta-function Z(C/Fp;T). For a typical hyperelliptic curve of genus g, the Katz-Sarnak model implies that this distribution matches the distribution of characteristic polynomials of random matrices in USp(2g). But there are many atypical cases: in genus 2 we already find 22 exceptional distributions.
I will describe the large scale numerical experiments (involving more than 1010 curves) that eventually led to a theoretical model that explains all of the exceptional distributions that have been observed in genus 2, and predicts that there are no others. Some key computational tools include: fast group operations in the Jacobian (borrowed from cryptography), and a method to quickly classify unknown distributions by approximating their moment sequences.
This is joint work with Kiran Kedlaya.
Yuri Tschinkel: Igusa integrals and volume asymptotics in analytic and adelic geometry (joint work with A. Chambert-Loir)
I will explain how to estimate volumes of height balls in analytic varieties over local fields and in adelic points of algebraic varieties over number fields, relating the Mellin transforms of height functions to Igusa integrals and to global geometric invariants of the underlying variety. In the adelic setting, this involves the construction of general Tamagawa measures.
Trevor Wooley: Rational points, Fermat hypersurfaces and mean values of Weyl sums
We report on work joint with Per Salberger that provides estimates for the number of rational points on complete intersections having degree large compared to the dimension. Such estimates may be applied in bounding the number of rational points on Fermat hypersurfaces in bounded domains, in estimating mean values of Weyl sums, and in allied problems concerning quasi-diagonal behaviour.
Don Zagier: Rational points and special L-values
Gordon Heier: On uniformly effective boundedness of Shafarevich Conjecture-type
The talk deals with uniformly effective versions of the classical Shafarevich Conjecture over function fields (aka Parshin-Arakelov Theorem). We will discuss the speaker's effective solution to the classical case and his recent extension to the case where the fibers are canonically polarized compact complex manifolds. In the proofs, Chow varieties play a key role.
Rafael von Känel: An effective Shafarevich theorem for elliptic curves
Let K be a number field and let S be a finite set of places of K. A classical theorem of Shafarevich says that there are only finitely many K-isomorphism classes of elliptic curves over K with good reduction outside S. An effective version of this statement for K=Q was already proved by Coates. In the talk we discuss an extension to arbitrary number fields. We give explicit bounds and compare them with the one obtained by Coates. This is joint work with Clemens Fuchs and Gisbert Wüstholz.
Jan Steffen Müller: Computing canonical heights on Jacobians
The canonical height is a quadratic form on abelian varieties defined over number fields that is an important tool in the study of their arithmetic. For instance, it can be used to determine explicit generators of the Mordell-Weil group. We will discuss how the canonical height can be computed in practice in the case of Jacobian varieties of curves using local methods.
Samir Siksek: The generalized Fermat equation x3+y4+z5=0
This talk is based on joint work with Michael Stoll (Bayreuth). The equation of the title was suggested by Zagier as the next case of the Generalized Fermat Conjecture. Work of Edwards reduces this equation to the determination of rational points on 49 hyperelliptic curves of genus 14. Standard methods for determining the rational points fail on many of these curves. We describe a new technique which we call 'partial descent' that succeeds in completing the determination of rational points on these curves. We deduce that the only solutions in coprime integers x, y, z satisfy xyz=0.
Sir Peter Swinnerton-Dyer: Density of rational points on certain K3 surfaces
Let V be a K3 surface containing at least two elliptic pencils. This talk discusses the density of rational points on V under the real and the p-adic topologies.
Szabolcs Tengely: On the Diophantine equation x2+C=2yn
In this talk we study the Diophantine equation x2+C=2yn in positive integers x,y with gcd(x,y)=1. We show how to obtain finiteness results for certain values of C, we also discuss bounds for the exponent n. The talk is based on a paper by Tengely and on a paper by Abu Muriefah, Luca, Siksek, Tengely.
Damiano Testa: Relative Weil restriction
We study an (easy) generalization of the Weil restriction functor to a relative setting and compute it for a morphism of curves. Combining this computation with either Faltings' Theorem or Chabauty's method we prove that certain sets of rational points on curves over number fields are finite.
As a sample application, we prove the following. Let L be a number field and let K be a subfield of L; suppose that f(x) is a square-free polynomial of degree four over L. If there are infinitely many solutions (x,y) in K x L to the equation y2=f(x), then there is a non-zero a in L such that af(x) is a polynomial with coefficients in K.
This talk is based on joint work in progress with E. V. Flynn.
Anthony Várilly-Alvarado: Transcendental obstructions to weak approximation on general K3 surfaces
It is well-known that K3 surfaces over number fields need not satisfy the Hasse principle or weak approximation. All known counter-examples to date, however, involve K3 surfaces that are endowed with an elliptic fibration structure; in fact, the fibration is essential to the computation of Brauer classes that reveal obstructions to the Hasse principle and weak approximation. General K3 surfaces, i.e., K3 surfaces with geometric Picard rank one, do not enjoy this kind of structure. I will explain how to construct certain K3 surfaces of geometric Picard rank one, together with a transcendental quaternion algebra that obstructs weak approximation of rational points. This is joint work with Brendan Hassett and Patrick Várilly-Alvarado.