2 INTRODUCTIONhand, Bernstein's lectures, which unfortunately are not included in this volume,
addressed the theory of Eisenstein series and their meromorphic continuation via
spectral theory and a principle of uniqueness. (For the case that the Eisenstein se-
ries are induced from cuspidal data on the Levi subgroup, a treatment along similar
such lines is due to Selberg and can be found in S. Wong [Memoirs of the A.M.S. Vol.
83 (1990)].) Lectures on Converse Theorems and the Langlands-Shahidi method
were delivered by Cogdell and Shahidi and their lecture notes are included here.
One of the significant consequences of the recent results on functoriality are new
bounds towards the Ramanujan and Selberg Conjectures for GL(2) due to Kim,
Sarnak and Shahidi. The Conjectures can be formulated quite generally for any
connected reductive algebraic group over a number field or a function field. The
basic assertion is that the local components of cuspidal representations on adelic
points of these groups must be tempered, i.e., they must have matrix coefficients
which li e in £^2 +< (Zc(Fv)\G(Fv)) for any E > 0, where Fv is a local completion
of F. Here Zc is the center of G. For GL(n) over function fields the conjectures
are a consequence of Lafforgue's proof of Langlands' global reciprocity Conjecture;
however, for a number field F nothing that strong is known. What is known are
quite sharp approximations. For GL(n) over a number field this is due to Rudnick,
Luo and Sarnak who use methods from the analytic theory of £-functions (specif-
ically Rankin-Selberg £-functions). Many of the new bounds obtained for cusp
forms on GL(2) and for other reductive groups are a consequence of the GL(n)
bounds and transfer principles. There are certain instances for which the con-
jecture is fully established such as that by Deligne for classical holomorphic cusp
forms on the upper half-plane and certain cases of Hilbert modular forms and uni-
tary groups (Harris, Taylor Blasius). These conjectures were the topic of Clozel's
Lectures which are included here. Besides the cases mentioned above he also ad-
dresses the connection with Arthur's conjectures on the non-tempered spectrum
in L^2 (Zc(A.F)G(F)\G(A.F)) and the subgroup, induction/restriction, technique of
Burger-Li-Sarnak giving global information from local calculations.
In the case of definite quaternion algebras the now proven Ramanujan Con-
jectures have applications to combinatorics and computer science through the con-
struction of Ramanujan Graphs (these are optimally highly connected but sparse
graphs called "expanders"). Other applications of this property of large spectral
gaps are to the counting and equi-distribution of integral points on homogeneous
spaces. The lectures by Li which are included in the volume cover these topics as
well as such higher dimensional combinatorial objects called Ramanujan buildings.
The lectures by Valette also discussed the closely related notion of property T for
groups.
The techniques of studying averages over "families of a utomorphic £-functions"
have proven to be very powerful in applications, particularly for establishing sub-
convex estimates for automorphic £-functions on their critical line as well as for
establishing non-vanishing results. For GL 2 /Q the sub-convexity was pioneered by
Duke-Fieldlander and Iwaniec and especially important is their method of amplifica-
tion. There are plenty of appli cations through special value formulae for £-functions
at the central point. These include a solution of Hilbert's eleventh problem on rep-
resentations of integers by quadratic forms over number fields (Cogdell-Piatetski-
Shapiro- Sarnak), ranks of Jacobians of modular curves .... all this and more are
discussed in the detailed notes by Michel which are included.