American Institute of Mathematics, Palo Alto, California
Matthew Greenberg, Markus Kirschmer, and John Voight
This workshop, sponsored by AIM and the NSF, will be devoted to explicit methods for algebraic modular forms.
In the late 1990s, Gross made a careful study of automorphic forms on reductive algebraic groups over the rational numbers with the property that every arithmetic subgroup is finite, dubbing these algebraic modular forms. Many split groups have inner forms with this remarkable property; the prototypical example is the group of units in the algebra over the rationals defined by Hamilton's quaternions, an inner form of $GL_2$ over the rationals. By the Langlands philosophy, algebraic automorphic forms give an explicit realization of automorphic forms on higher rank groups that is particularly amenable to computation. When computing spaces of algebraic modular forms, the main workhorse is a suite of algorithms for lattices, specifically algorithms for isomorphism testing of lattices that respect a positive definite quadratic form.
The goal of this workshop is to bring together researchers in number theory, arithmetic geometry, algebraic groups, and lattices--in both their theoretical and computational aspects--to lay the foundation for general methods of computing spaces of algebraic modular forms for a large class of reductive algebraic groups.
The main topics for the workshop are:
The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.
The deadline to apply for support to participate in this workshop has passed.
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