Modular Forms: Arithmetic and Computation (07w5065)

Arriving in Banff, Alberta on Sunday, June 3 and departing Friday June 8, 2007

Organizers

John Cremona (University of Nottingham)
Henri Darmon (McGill University)
Kenneth Ribet (University of California at Berkeley)
Romyar Sharifi (University of Arizona)
William Stein (University of Washington)

Objectives

The objectives of our proposed workshop fall roughly into three categories:

1) to bring together leading arithmetic geometers working on various aspects of modular forms and closely related objects, with the purpose of informing each other of recent groundbreaking developments,

2) to explore the relationship between computation and theory within the context of the study of modular forms, and

3) to facilitate new collaborations and mutually beneficial interaction between arithmetic geometers of various degrees of computational inclination.

Our workshop is concerned with modular forms and the objects to which they are intricately connected, such as elliptic curves, p-adic L-functions, and Galois representations. We will maintain a sharp focus on actual modular forms, as opposed to the broad variety of analogous objects. Even with this focus, the topic is broad enough to encompass a great deal of the most exciting work in arithmetic geometry today.

In the last few years, there has been a tremendous amount of groundbreaking progress on open questions relating to modular forms. Much of this has been detailed in the attached overview. The work of Khare and Khare-Wintenberger on Serre's conjecture, the work of Kato and Skinner-Urban on the main conjecture for modular forms, and Kisin's work on the modularity of Galois representations are among the most fascinating of many recent advances. Our workshop provides an excellent opportunity to bring together these and other researchers to inform each other on the details of major developments in the field. It also serves as a valuable forum for the education and advancement of younger researchers, a number of whom we intend to invite, and many of whom will be at the forefront of forthcoming developments. Finally, but not least of all, it acts as a catalyst for the formation of new collaborations among the participants.

A second focus of the workshop will be the computational side of the study of modular forms and its interaction with theory. Since the early days of computers, computation has played a crucial role in the formulation of many of the most important conjectures involving modular forms. (Again, this is partially described in the overview.) The ability to compute with modular forms has made the job of modern theoreticians substantially easier, for they no longer need suffer from a paucity of examples upon which to base their conjectures. Today, a wealth of computational data and algorithms for modular forms and elliptic curves exists, thanks in large part to the efforts of John Cremona and William Stein. Numerous routines are now available for computation with these objects in computer algebra packages such as Magma, PARI, and SAGE. It is possible to use computers for theoretical purposes that were nearly unimaginable a quarter of a century ago.

Indeed, computing is becoming an increasingly useful and prevalent tool in the number theorist's arsenal. Nevertheless, and rightly so, modern arithmetic geometers vary greatly in the use of computers in their research. The participants will represent a wide range of styles in their approach to computation in mathematics, and we believe this variety will be of benefit to all. Those who are not at the forefront of developing algorithms for modular forms will have the opportunity to learn of the tools, packages, and algorithms available to them and to form collaborations with the computationally-minded for the purpose of developing theory through computation. Conversely, those developing algorithms will benefit from hearing what other arithmetic geometers are interested in being able to compute. The resulting interactions will be of tremendous potential benefit to the field.

We should point out that the subject of this workshop is of an entirely distinct nature from any other 5-day workshop offered in Banff during the years 2004-2006. On the other hand, its topic of modular forms lies dead center in modern arithmetic geometry. This enables us to bring together some of the most well-established and respected arithmetic geometers in the world, along with some of the best younger mathematicians in the field. With the wide range of important recent developments on open questions in the field and the ever increasing availability of computational algorithms and data, we feel that it is a perfect time to hold this workshop.