Rational Points on Curves and Higher Dimensional Varieties

Monday 16th June - Friday 20th June 2008

 

This school is part of the April–June Active Period in Arithmetic Geometry and the 2007-2008 Warwick Maths Institute EPSRC Symposium.

Background

Suppose X is a curve defined over Q with X(Q) non-empty. Then it known that X(Q) is either parametrizable or a finitely-generated abelian group or a finite set according to whether the genus of X is 0, 1, or at least 2. This is an example of how geometry determines arithmetic.

Yet, there is still no algorithm, which given a curve X of genus at least 1, is guaranteed to determine whether X(Q) is non-empty. Moreover, even if X(Q) is non-empty, there is no known algorithm guaranteed to determine the basis of the Mordell-Weil group in genus 1, or the set itself in genus 2. Despite this seemingly bleak outlook, the last few years have seen the development and refinement of many viable strategies, which although not guaranteed to answer these questions in each instance, are likely to be successful in doing so.

For varieties of higher dimension, much less is known about how the geometry determines the arithmetic, although this has become a subject of intensive research. Here experimentation is likely to provide conjectural partial answers to this fundamental question.

Aims

We bring together many leading experts in the subject and aim to give them an opportunity to present their latest results and insights. To create a situation that leads to fruitful exchange, the participants include both people working on the theoretical side of the subject, and those working on the explicit or experimental side. Through this we hope to achieve a better understanding of the current state of the art and to provide the basis for animated discussions. More importantly, we aim to identify and explore the most promising directions for future work.

Format

The conference will start on Monday morning and finish on Friday at about midday. There will be at most four lectures per day, to give the participants enough time for discussions and collaborations.

Visit the Workshop Website for more information on this event.

Registration is required in advance for this event

See the main Symposium website for details of the symposium, including a calendar of all events occurring and how to register for a place on any course or workshop.