FOREWORD 183
sufficient (mainly because they are involutory) and so they must be sup-
plemented by various techniques of transformation or of estimation, some
of which will be described in the following lectures.
- In the third lecture, we discuss an important method involving families:
the large sieve method and the large sieve type inequalities. These in-
equalities are relatively coarse but also very robust and they have many
interesting applications, one of which is the problem of bounding non-
trivially the dimension of the space of holomorphic forms of weight one
and given conductor. By an elegant application of the large sieve, Duke
was the first to make progress on this question. We also show how such
inequalities can be used to produce zero density estimates that are good
substitutes for GRH.
- The fourth lecture is the most significant: it gives an overview of the
various ingredients used in the resolution of the Subconvexity Problem
of GL 1 , GL 2 and CL 2 x GL 2 , £-functions which is by now to a large
extend solved.
(1) The methods of Weyl and Burgess work well for £-functions of de-
gree one but are hardly extendible to £-functions of higher degree.
(2) The method of moments and the amplification method, which are
built on families, are to date the most general methods available
for solving the Subconvexity Problem. The amplification method
may have other applications: for instance, we will use it to give
easy improved bounds for the dimension of the space of holomorphic
forms of weight one.
(3) Next, we show how these methods reduce the Subconvexity Prob-
lem for modular £-functions to another one: the Shifted Convo-
lution Problem, SCP. It consists of bounding non-trivially partial
sums of Rankin/ Selberg type, but with an additional non-trivial ad-
ditive twist. We describe two somewhat independant methods for
solving the ScP (in fact, these methods are related via the Peters-
son/Kuznetzov trace formula): an elementary approach that builds
on the 8-symbol and relies ultimately on non-trivial bounds for
Kloosterman sums, and a spectral approach, inspired by the Rankin-
Selberg unfolding method.
( 4) The latter approach uses the full force of the theory of Maass forms
(even if one is only interrested in £-functions of holomorphic forms)
and requires a non-trivial bound for their local parameter at infin-
ity. It also depends on good bounds for integrals of triple products
of modular forms: such bounds can be obtained by various rather
advanced techniques, which we have no time to describe here.
Finally, as an illustration, we collect all these methods together in the
proof of the Subconvexity Problem for Rankin-Selberg £-functions (which
are of degree 4). Interestingly, this case "closes the circle," since the proof
(of this individual estimate), starts with families and ends up after sev-
eral transformations with another set of non-trivial individual estimates
for another set of modular forms (of course quite different from the one
we started from): namely the non-trivial approximations to the RPC for
Maass forms and the non-trivial (subconvex) bounds for £-functions of
