Rational Points on Curves and Higher Dimensional Varieties

Monday 16--Friday 20 June 2008

at the

Mathematics Research Centre, University of Warwick, UK

organized by

Jean-Louis Colliot-Thélène, Samir Siksek, Alexei Skorobogatov, Michael Stoll

Programme (pdf)


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.


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.


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.


Maria Aranes (Warwick), Carlos Barros (Warwick), Alex Bartel (Cambridge), Martha Bernal (Warwick), Nicolas Billerey (jussieu), Martin Bright (Bristol), Tim Browning (Bristol), Nils Bruin (Simon Fraser), Jean-Louis Colliot-Thélène (Orsay), Stephen Coughlan (Warwick), John Cremona (Warwick), Brendan Creutz (Bremen), Sarah Davis (Warwick), Cyril Demarche (Orsay), Jamshid Derakhshan (Oxford), Ulrich Derenthal (Zurich), Tim Dokchitser (Cambridge), Noam Elkies (Harvard), Tom Fisher (Cambridge), Victor Flynn (Oxford), Homero Gallegos (Warwick), Jean Gillibert (Bordeaux), David Harari (Orsay), Michael Harrison (Sydney), William Hart (Warwick), Michael Harvey (Bristol), Umar Hayat (Warwick), Roger Heath-Brown (Oxford), Mostafa Ibrahim (Warwick), Evis Ieronymou (Imperial), Sohail Iqbal (Warwick), Kamal Khuri-Makdisi (American University Beirut), Minhyong Kim (University College London), Ronald van Luijk (Warwick), Odile Lecacheux (Paris 6), Daniel Loughran (Bristol), Valery Mahe (East Anglia), Gihan Marasingha (Oxford), Tzanko Matev (Bremen), Anna Morra (Bordeaux), Jan Mueller (Bremen), Nic Niedermowwe (Oxford), Ambrus Pal (Imperial), Emmanuel Peyre (Université Joseph Fourier), Bjorn Poonen (Berkeley), Thomas Preu (Zurich), Miles Reid (Warwick), Mohammad Sadek (Cambridge), Edward Schaefer (Santa Clara), Samir Siksek (Warwick), Graham Sills (Cambridge), Denis Simon (Caen), Alexei Skorobogatov (Imperial), Michael Stoll (Bremen), Peter Swinnerton-Dyer (Cambridge), Damiano Testa (Bremen), Thotsaphon Thongjunthug (Warwick), Jan Tuitman (Leuven), Yukihiro Uchida (Nagoya), Tony Varilly (Berkeley), Bianca Viray (Berkeley), Mark Watkins (Bristol), Olivier Wittenberg (Strasbourg), Trevor Wooley (Bristol), Christian Wuthrich (Nottingham), Andrei Yafaev (University College London) Shengtian Zhou (Warwick).

Registration is now closed.

This event is part of the April–June Active Period in Arithmetic Geometry and the 2007–2008 Warwick EPSRC Symposium on Algebraic Geometry