184 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS
degree 1 and 2 (which have been proved before). Thus one may suspect
that the complete use of the interplay between individual type bounds
and averaged bounds is far from finished.- In the fifth and last lecture we discuss many applications of the Subcon-
vexity Problem. Our aim is to convince the reader, by means of examples,
that a subconvex bound is not simply another improvement over some ex-
isting exponent, but has an intrisic geometrical or arithmetical meaning.
For instance, we show how in several cases the convexity bound matches
exactly another bound obtained by other generic methods of geometric or
arithmetic nature (the Riemann/Roch or the Minkowski Theorem). Sub-
convexity bounds are also used to establish a variety of equidistribution
results, ranging from the distribution of lattice points on the sphere, to
the Heegner points, and the context of Quantum Chaos.
There are several important topics connected with families of £-functions that can
be handled by similar techniques and which, for lack of time and energy, we have
not treated in these lectures. One is the problem of proving the existence of an £-
function in a given family that does not vanish at some special (meaningful) point.
An example however, is given at the end of the first lecture. This kind of prob-
lem can be handled by many methods (such as the mollification method) and has
many applications in various fields. The other topic is the "Katz/Sarnak philoso-
phy," which is a net of far reaching conjectures (going far beyond GRH) describing
the local distribution of zeros of families of £-functions in terms of the eigenvalues
of random matrices of large rank. Although this field is highly conjectural, it is nev-
ertheless important, as it reveals beautiful inner structures in families and provides
a unified framework for various phenomena occurring in analytic number theory;
moreover, it strongly suggests that further progress on GRH will come from the use
of families.
These notes are an extended version of a series of five lectures given during
the Park City Mathematical Institute in july 20022. I hope that these notes have re-
tained the informal style of the lectures. In particular, few proofs have been given in
full detail: we hope that this will serve to capture better the main ideas. To fill the
many remaining gaps, the reader will need to look further at the existing literature.
The other lectures of the present volume should be fully sufficient to cover the part
relevant to the general theory of automorphic forms. For a more complete intro-
duction to the general methods of analytic number theory, several good books are
available: for the basics, the reader may consult Davenport's "Multiplicative Num-
ber Theory" and Tenenbaum's "Introduction a la Theorie Analytique et Probabiliste
des Nombres" (also available in english) and for an introduction to advanced topics
Bombieri's little (big) book "Le Grand Crible clans la Theorie Analityque des Nom-
bres". For us, the ultimate reference is Kowalski/Iwaniec's book "Analytic Number
(^2) ADDED IN PROOF (2006): since then, there has been a lot of progress made regarding several
topics discussed in these lectures. One of the most striking recent progress, in our opinion, are the
Berstein/Reznikov and Venkatesh new approaches to the subconvexity problem [Ve, BR3]. Their meth-
ods differ from the ones discussed in these lectures in that the subconvexity problem is approached in
terms of bounds for periods of automorphic forms rather than of bounds for central values of £-functions
and this has many advantages. For instance, in [Ve], many cases of the subconvexity problem in the
level aspec are proven for £-functions related to GL 1 and GL 2 automorphic representation over an
arbitrary fixed number field!