One of the remarkable things about this theorem is the way in which it suggests that geometry informs arithmetic. The geometric genus g is a manifestly geometric condition, yet it is controlling what seems to be an arithmetic property. Why should the number of integral solutions to xn + yn = zn have anything to do with the shape of the complex solutions? You might argue that that the genus is e...

Let k be an algebraic number eld and F (x0; x1; x2; x3) a non{singular cubic form with coeecients in k. Suppose that the pro-jective cubic k{surface X P 3 k given by F = 0 contains three coplanar lines deened over k, and let U (k) be the set of k{points on X which does not lie on any line on X. We show that the number of points in U (k), with height at most B, is OF;"(B 4=3+") for any " > 0.

Introduction 1. Comparison of Density Conditions 2. Curves 3. Conic Bundles over P 4. Smooth Cubic Hypersurfaces 5. Smooth Complete Intersections of Two Quadrics in P 6. Elliptic Surfaces 7. Some Abelian Varieties 8. Some Kummer Surfaces 9. Some Other K3 Surface Examples References The aim of this article is to provoke a discussion concerning the general nature of the topological closure of the...

A classical theme in mathematics is the study of integral solutions of diophantine equations, that is, equations with integral coefficients. The main problems are – decide the existence (or nonexistence) of solutions; – find (some or all) solutions; – describe (qualitatively or quantitatively) the set of solutions. Even the first of these questions is difficult, in full generality. As we know f...

The purpose of this talk is to put forward a conjecture. The background is given by the following Basic Question. Given a (smooth projective) curve C over a number field k, can we determine explicitly the set C(k) of rational points? One possible approach to this is to consider an unramified covering D π → C that is geometrically Galois. By standard theory, there are only finitely many twists D...

Epistemic justification has been claimed to be a special case of argument by philosophers such as Accordingly for each attempt, epistemic justification is the occasion of the arguer making a case for some conclusion based on other premises, when the audience and the arguer are the same person. The dialectical model of argument is used to inform how it is that a subject could come to justify her...

x.1. Elliptic Surfaces Deenition. An elliptic surface consists of a smooth (projective) surface E, a smooth (projective) curve C, and a morphism : E ?! C such that almost all bers E t = ?1 (t) are (smooth projective) curves of genus 1. In addition, we will generally assume that our elliptic surfaces come equipped with an identity section 0 : C ?! E which serves as the identity element of the gr...

The Lanczos method and its variants can be used to solve eeciently the rational interpolation problem. In this paper we present a suitable fast modiication of a general look-ahed version of the Lanczos process in order to deal with polynomials expressed in the Chebyshev orthogonal basis. The proposed approach is particularly suited for rational interpolation at Chebyshev points, that is, at the...