242 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS
and the proof of the s aspect for general Dirichlet L-functions is similar. However,
very hard work has been done to improve Weyl's exponent of 1/6; the current
record (to date), 32/205, is held by M. Huxley [Hu].
The next important progress is due to Burgess [Bu] who, in the 60's, solved the
Subconvexity Problem for Dirichlet L-functions in the q-aspect: one has
(4.2) L(x, s) «E lslAq~l^1 6+E,
for some absolute constant A. The exponent 3/16 resisted any improvement for the
next 40 years, until the breakthrough of Conrey/ Iwaniec [Conl] in the case of real
characters: one has for any c > 0
(4.3) L(x, s ) «E lslAq~/6+E.
It is remarkable that this exponent (which matches Weyl's original exponent for()
required the full power of the spectral theory of GL 2 , positivity results for central
values of automorphic L-functions due to Waldspurger and others, as well as the
Weil conjectures over finite fields.
Things accelerated in the 80's with the first subconvexity bounds in the s-aspect
for GL 2 L-functions, arising from the works of Good [Go] and Meurman [Me].
The spectral aspect for Hecke L-functions was treated by Iwaniec [13] - who in-
troduced on this occasion the fundamental amplification method-which we will
describe below, and by Ivie, Jutila, and Peng [Iv2, Ju4, Pe]. The level aspect for
G L 2 L-functions was treated by Duke/ Friedlander/ Iwaniec in a series of papers
stretched over the 90's ([DFII]. .. [DFI8]) culminating with the very difficult:
Theorem 4.1. Let f E S~(q,x,it). Assume that xis primitive, then there exists a
positive b(;? 1/24000) such that
L(f, s) «k,it,s q^1!^4 -o ·
And recently the case of Rankin-Selberg L-functions has been treated, by Sar-
nak and Liu/Ye for the spectral aspect and Kowalski/ Michel/Vanderkam and Michel
for the level aspect [Sa4, LY, Kl\/IV2, Mi].
In the forthcoming sections, we present the various techniques that can be used
to obtain the above subconvexity theorem in all cases. We only discuss the q-aspect
here but we emphasize that these techniques apply equally to the other aspects
with appropriate modifications.
We end this etat des lieux with the case of more general number fields: several
instances of the Subconvexity Problem for GL 2 L-functions have been solved by
Petridis/ Sarnak and Cogdell/Piatetsky-Shapiro/ Sarnak [PS, CoPSS]. We highlight
the latter work, as it is the key to the final solution of Hilbert's 11th problem on the
representation of integers by quadratic forms in number fields^1 (see [Col]):
Theorem 4.2. Let F be a totally real number field, Xq a ray class character of modulus
q, and g a holomorphic Hilbert modular form over F. One has for any c > 0
L(g.xq' s) «E,g N K / Q ( q)1;2-1 ; 130+£
(^1) Added in proofs: in a recent preprint dated of June 2005, A Venkatesh, by a method very different
of the one presented here, generalized the bound of Theorem 4 .2 to arbitrary GL 2 -automorphic forms
over any number field and in fact generalized to a fixed arbitrary number field much of the subconvex
bounds in the level aspect presented above [Ve]