MSRI Modular Forms Summer Workshop Problem Book

Organizer: William Stein

Date: July 31-Aug 12, 2006

Abstract:

This MSRI workshop will be focused on making progress toward the problems listed in this document. The goal of the workshop will be to produce papers, software, and web pages related to each of the problems. These documents should be of interest to active researchers in modular forms (target audience: the workshop speakers, and, e.g., Barry Mazur, Karl Rubin, Ralph Greenberg, Matt Emerton, Frank Calegari, graduate students, etc.)

Instead of dividing the participants into groups with leaders the five invited lecturers, we will instead be dividing into groups who will attack 8 groups of research problems. These problems are not meant to span the whole gambit of problems about computing with modular forms - they are a necessarily biased selection of problems.

Everybody should find their email address listed in the group for at least two problems below. (If not, email [email protected].) You should feel fine focusing primarily on the problems you've been assigned to. You'll probably discover that you make progress on one problem, but not the other, which is perfectly fine - that's why you have two problems. Also, everybody should feel welcome to look at or help with any problem.

Do not be unduly discouraged if you find every problem difficult - they are unsolved problems, unless otherwise noted. Many could lead to a Ph.D. thesis.

Most problems below that say ``implement in SAGE'' can be interpreted to mean ``implement using only free open source software,'' since it is very easy for SAGE to include any such programs. Moreover, by ``algorithm'' below we usually mean a ``practical algorithm'', i.e., something that could be implemented and actually used to obtain interesting insight.

There is also a final ``fun'' chapter below consisting entirely of questions that ask for ways to draw pictures of mathematical objects attached to modular forms.