Français >


The explosive development of the theory of automorphic forms on GL(2) in the second half of the twentieth century allowed a number of spectacular recent advances in number theory.  The most visible of these was the proof of the Shimura-Taniyama-Weil conjecture (and of Fermat's Last Theorem) almost 15 years ago, but one should also mention other highlights like the subsequent proof of the the Serre conjectures relating two-dimensional mod p Galois representations to modular forms, and the important progress on the Fontaine-Mazur conjectures concerning p-adic Galois representations.Spurred in part by some of these breakthroughs, the last decade has witnessed significant arithmetic applications of the  theory of automorphic forms on higher rank groups, such as the spectacular recent proof of the Sato-Tate conjecture and the ongoing work on the Iwasawa main conjecture exploiting Eisenstein series on unitary groups. Automorphic forms on GL(2) are very accessible to machine calculation, and there has been a tremendous amount of activity devoted to developing efficient algorithms for computing with modular forms, and making tables of  data related to modular forms widely accessible.  For example, we now have extensive on-line tables of  all elliptic curves over Q of conductor up to 70000, a database which contains several million elliptic curves and whose calculation, based on the truth of the  Shimura-Taniyama conjecture and on the modular symbol method for computing holomorphic cusp forms, is both a computational tour de force and a real treasure trove for researchers interested in the arithmetic of elliptic curves. By contrast, the realm of automorphic forms on semisimple groups of higher rank (and even, on inner forms of GL(2)) has been much less explored computationally, and this presents a great number of interesting  challenges.  It would be of great interest, both for theoretical as well as experimental purposes, to  bring the theory of automorphic forms as much as possible within reach of practical machine calculations, and the ASI that we are proposing is informed by this desire. The purpose of the ASI is to bring together a number of experts who have been leaders in both the theoretical and more computational aspects of the theory of automorphic forms, who will offer introductory-level courses aimed at presenting graduate students and post-doctoral fellows with the state of the art in the subject and report on new advances which have not yet been covered in a forum of this sort.