Classical spectral decomposition, both local and global, are introduced. In particular, when applied to the general linear group $GL_r$, this leads to the local and global $L$ and $\varepsilon$-factors, once we enlarge the base space from $GL_r$ to $M_r$, the space of $r\times r$-matrices.

Recorded from 10:10am of April 26, 2013, this is the first talk given at the symposium on 'Automorphic Functions and Arithmetic Geometry', held at the Graduate School of Mathematics, Kyushu University, Fukuoka, Japan.

Local and global $L$ and $\varepsilon$-factors are introduced and their standard properties are summarized. The results are classical, but the way they are presented here is original.

Recorded from 11:10am of April 26, 2013, this is the 2th talk given at the symposium on 'Automorphic Functions and Arithmetic Geometry', held at the Graduate School of Mathematics, Kyushu University, Fukuoka, Japan.

Langlands' automorphic transfer conjecture and its consequences are presented. The transfer rules by a transfer representation $\rho$ proposed by Langlands lead to the definition of non linear $L$-functions, whose global propoerties are deduced from the automorphic transfer conjecture.

Recorded from 10:10am of April 27, 2013, this is the 3th talk given at the symposium on 'Automorphic Functions and Arithmetic Geometry', held at the Graduate School of Mathematics, Kyushu University, Fukuoka, Japan.

New types of function spaces and of $(\rho,L)$-Fourier transforms on these spaces are associated to transfer representations $\rho$, through local spectral decomposition. Non-linear Poisson formulas without boundary terms are presented and are deduced from Langlands' automorphic transfer conjecture.

Recorded from 11:10am of April 27, 2013, this is the 4th talk given at the symposium on 'Automorphic Functions and Arithmetic Geometry', held at the Graduate School of Mathematics, Kyushu University, Fukuoka, Japan.

The non-linear Poisson formulas without boundary terms introduced in the previous lecture are conjecturally generalised to non-linear Poisson formulas with boundary terms.

Recorded from 10:10am of April 28, 2013, this is the 5th talk given at the symposium on 'Automorphic Functions and Arithmetic Geometry', held at the Graduate School of Mathematics, Kyushu University, Fukuoka, Japan.

The notion of 'kernels' of Langlands' automorphic tranfer is introduced. As an application, Langlands' automorphic tranfer conjecture is deduced from the non-linear Poisson formulas.

Recorded from 11:00am of April 28, 2013, this is the last talk given at the symposium on 'Automorphic Functions and Arithmetic Geometry', held at the Graduate School of Mathematics, Kyushu University, Fukuoka, Japan.

This talk presents the Langlands programme as a set of conjectures linking together two (or even three) mathematical theories which are a priori completely different:

(1) the theory of the finite separable extensions of a base field which happens to be a global field,

(2) the conjectural theory of pure motives over such a global field, related to the previous theory through the l-adic cohomology functors,

(3) the theory of automorphic representations of linear groups (or more generally reductive groups) over such a field.

Recorded from 16:00, May 1, 2013, this is the talk of Prof. Lafforgue given at the First Kyushu Joint Seminar, held at the Graduate School of Mathematics, Kyushu University, Fukuoka, Japan.