SMS 2009

Automorphic Forms and L-Functions: Computational Aspects

June 22 - July 3, 2009
Département de Mathématiques
Université de Montréal

Historic Background.  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 130000, 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.

Purpose.  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.

Timeliness and Importance.  The topics of the ASI are enjoying a worldwide surge of interest spurred by two related developments. On the theoretical side, one can mention the significant arithmetic applications of the theory of automorphic forms on higher rank groups to questions like the Sato-Tate conjecture and the Iwasawa Main Conjectures.  On the practical side, the last decade has seen the development of powerful software for number theoretic calculations, the SAGE package being a recent and striking example, as well as the emergence of a burgeoning and increasingly important "experimental community" within the number theory community.  Furthermore,  the computational aspects of number theory and modular forms play an essential role in a growing number of  cryptographic protocols, in the construction of expanders, and in the design of good  error-correcting codes.

For the past few years, there have been several meetings devoted exclusively to algorithmic number theory with emphasis on calculations related to modular forms, most notably the yearly ANTS meetings, and a meeting last summer at the Banff Center. The ASI that we propose to put together distinguishes itself from these gatherings in that is aimed at graduate students, and therefore has an important instructional component. Because the subject of computations with modular forms is so recent, and has developed so rapidly in the last years, there are few textbooks that  are available at that level. Perhaps the only one is the textbook "Modular forms, a computational approach" by William Stein, who is one of the  leaders of this young subject.  The proceedings of our conference will cover material that is somewhat more specialised and at the frontier of current research, so we anticipate little overlap with Stein's more basic and foundational textbook.

Applications and relevance for the Research Topics of Special Interest: One of the purposes of the ASI will be to present the state of the art in algorithmic number theory, a rapidly developing field that has greatly benefitted from the explosive growth of symbolic algebra  software. A potential side benefit of the ASI will be to increase awareness of the possibilities that these new technologies have created, and to expose young people to a research area at the interface of pure mathematics and computer science whose impact is sure to spill over to other areas as well such as scientific computation.

Titles and Speakers

- Andrew Booker, Maass forms and L-functions
- Henri Darmon and Eyal Goren, Background and prerequisites
- Noam Elkies, K3 surfaces of high Picard number and their moduli
- Dorian Goldfeld, Automorphic forms and L-functions in higher rank
- Kamal Khuri-Makdisi, Modular interpretation and equations for modular varieties
- Erez LapidThe Arthur-Selberg trace formula and applications
- Michael Rubinstein, Algorithms for the computation of L-functions
- Harold Stark, Artin L-functions and special values at s=0
- Fredrik Strömberg, Maass waveforms for SL(2,Z) and subgroups, from a computational point of view
- Audrey Terras, L-functions of Graphs
- Andreas Strömbergsson, Computations using the Selberg Trace Formula
- Akshay Venkatesh, Eisenstein series
- John Voight, The algorithmic theory of quaternion algebras

Scientific Directors

Michael Rubinstein (Waterloo)

Andreas Strömbergsson (Uppsala)

Organizing Committee

Mike Rubinstein (Waterloo)
Andreas Strömbergsson (Uppsala)
Henri Darmon (McGill)
Eyal Goren (McGill)