# Algorithms for lattices and algebraic automorphic forms

May 6 to May 10, 2013

at the

American Institute of Mathematics, Palo Alto, California

organized by

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:

1. The p-neighbors algorithm: How does it apply to an arbitrary reductive group (in terms of the building) and how is the general setup best described explicitly? In particular, how are the Hecke operators given by p-neighbors on a general group?
2. Lattice isomorphism testing, representations, mass formula: What is the full set of algorithmic methods, optimizations and improvements known? Can the extra structure of a lattice arising by restriction from a totally real field be harnessed to improve these algorithms? What is the most efficient way to compute with representation modules and the mass formula for a reductive group (encoding the weight and level)?
3. Theoretical running time estimates: Does the lattice method for computing algebraic modular forms run in polynomial time in the output size (Ramanujan-Petersson conjecture)? What are the exact relationships between the mass, the class number, the sizes of the automorphism groups, and the diameter of the neighbour graph on isometry classes?
One further goal of the workshop would be to initiate discussion and collaboration on open problems: what data should be computed, how would it be useful, and what are the interesting conjectural questions which need experimental evidence, where are the interesting Galois representations? The suite of algorithms that we seek to develop will be designed to help researchers develop insights into the Langlands program. In particular, since the algebraic modular forms make sense in any characteristic, the ability to compute with them should serve to guide and provide evidence for the conjectures of the nascent p-adic and mod p Langlands programs. Further, although lattices algorithms are important tools for the computational theory of automorphic forms, the connections described above work equally well in the other direction--explicit knowledge of spaces of automorphic forms gives information about the existence and structure of lattices.

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.