#### 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"

#### Abstracts:

#### Henri Darmon:

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.

#### Paul Gunnels:

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.

#### David Loeffler:

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.

#### Robert Pollack:

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.