Confirmed speakers of the summer school:
Henri Darmon (McGill University, Canada)
2 guest lectures on Thursday and/or Friday
Paul Gunnells (University of Massachusetts, USA)
Title: "Arithmetic groups, automorphic forms, and Hecke operators"
David Loeffler (Warwick University, United Kingdom)
Title: "Automorphic forms for definite unitary groups"
Robert Pollack (Boston University, USA)
Title: "Overconvergent modular symbols"
The two lectures I would like to give on Thursday and Friday will be entitled: "p-adic Rankin L-functions: a computational perspective." In this series, I will describe the construction of p-adic Rankin L-functions and raise the question of how to compute them efficiently in polynomial time.
Arithmetic groups are discrete subgroups of Lie groups; for basic
examples one should think of the modular group SL(2,Z), the Siegel
modular group Sp(2n,Z), and their congruence subgroups. The
cohomology of such groups provides a concrete realization of certain
automorphic forms, in particular automorphic forms that are
conjectured to have a close relationship with arithmetic geometry.
For instance, by results of Eichler-Shimura the cohomology of
congruence subgroups of the modular group gives a way to explicitly
compute with holomorphic modular forms.
In this course we will explore this connection between topology and
number theory, with the goal of presenting tools one can use to
compute with these objects. We will review the situation for SL(2,Z)
and then will discuss how one computes with other groups, especially
SL(n,Z) and GL(n) over number fields; the latter family includes
Hilbert modular forms. Special emphasis will be placed on how one can
compute the action of the Hecke operators on the cohomology
corresponding to cuspidal automorphic forms.
Topics for the lectures include the following: cohomology of
arithmetic groups and connections with representation theory and
automorphic forms, modular symbols, explicit polyhedral reduction
theory, Hecke operators, connections to Galois representations.
I will describe an approach to computing automorphic forms for a
certain class of reductive groups where the theory can be made purely
algebraic. The most prominent examples of such groups are definite
unitary groups (in any number of variables). I will explain the
construction (due to Gross) of algebraic automorphic forms for such
groups, and how this leads naturally to an algorithm for calculating
these spaces using lattice enumeration techniques. I will illustrate
this with some examples of computational results for unitary groups in
3 and 4 variables, and describe how the results can be interpreted in
terms of Galois representations.
The theory of modular symbols, which dates back to the 70s, allows one
to algebraically compute special values of L-series of modular forms.
In the 90s, Glenn Stevens introduced the notion of overconvergent
modular symbols which is a p-adic extension of the classical theory.
In the end, overconvergent modular symbols encode p-adic congruences
between special values of L-series, and in particular, are intimately
related to p-adic L-functions.
This course will give a down-to-earth introduction to the theory of
overconvergent modular symbols. This theory has the great feature of
being extremely concrete, and as a result, extremely computable. We
will explain concrete methods to compute overconvergent symbols in
practice, and as an application, one obtains algorithms to compute
p-adic L-functions of modular forms.
Last Update: 19.10.2011 - 16:05