182 PH. MICHEL, ANALYfIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS
of the last century was devoted to the development of techniques capable of provid-
ing good substitutes for unproved hypotheses (zero density theorems, for instance
are used as a substitute to GRH). Another (probably as important) motivation is
the recent realization that global analytic estimates for a family of £-functions can
be used to give non-trivial estimates for each of its members (or sometimes for
other related £-functions). The method of moments or its refinement, the amplifi-
cation technique, is able to provide sharp individual estimates for critical values of
£-functions and is a good example of this principle. A striking application of the
method of moments is the recent improvement by Conrey/ Iwaniec of Burgess's 40
years standing bound for the values of L (x, s), for x a quadratic character and s
on the critical line. Interestingly, the bound follows from a bound for the third mo-
ment of the central value £-function for a family of GL 2 -modular forms (including
Eisenstein series), rather than for a family of Dirichlet characters. The fact that in-
dividual estimates can be obtained from global ones is another (somewhat coarse)
manifestation of the powerful principle that much insight can be gained for an in-
dividual object, if one is able to deform it into a family and to obtain enough global
information on the deformation space: this principle was beautifully illustrated, in
the past, in Deligne's proof of the Weil conjectures (GRH for £-functions over finite
fields) and in the more recent solution by Wiles of Fermat's Last Theorem.
The lectures are organized as follows:
- In the first lecture, we review various aspects of the analytic theory of
individual automorphic £-functions; more precisely, we describe their
functional equation, the standard conjectures and what is known un-
conditionally: this includes the bounds for their local parameters (the
Ramanujan/Petersson Conjecture, RPC), the location of their zeros (the
Generalized Riemann Hypothesis, GRH), and the size of their values on
the critical line (the Generalized LindelOf Hypothesis, GLH). We prove the
so-called convexity bound, which is the best result towards GLH known in
general, introduce the Subconvexity Problem (ScP), which is the prob-
lem of improving the convexity bound, and describe how families can be
used to solve it. At the end of the lecture, we illustrate the usefulness of
families of automorphic forms for individual estimates with the Theorem
of Luo/Rudnick/Sarnak, which gives the best general (and non-trivial)
approximation to the RPC. - The second lecture gives a short description of the analytic theory of G L 2 -
automorphic forms from the classical viewpoint: so far, it is essentially in
the classical setting, that the most advanced aspects of the analytic the-
ory have been developed (a notable exception is the recent subconvex-
ity bound for £-functions for Hilbert modular forms of Cogdell/ Piateski-
Shapiro/Sarnak). We compare automorphic £-functions with their clas-
sical counterparts. We also give various "trace formulas," which are the
main tool for performing averaging over families; these formulas trans-
form an averaging over a family into a "dual" side that putatively is
more tracktable. Particularly important to the G L 2 theory is the Peters-
son/ Kuznetsov formula, which expresses the average of the Fourier coef-
ficients of a modular form in terms of sums of Kloosterman sums. Note
that for analytic purposes, this formula is more powerful than the Sel-
berg trace formula, for instance. However, trace formulas alone are not