Lecture 4. The subconvexity problem
In this lecture we describe the state of the art regarding the Subconvexity Prob-
lem (ScP) given in the first lecture; almost all of what is known about the problem
concerns essentially £-functions of GL 1 ,q and GL 2 ,q automorphic forms. Remem-
ber that there are basically three aspects to consider regarding the size of an £-
function associated to a modular form f, each of which one may let grow to +oo:
the level qi, or the spectral parameter lμi, 11 (which is essentially (k1 - 1)/2 if f
is holomorphic, or the spectral parameter lit I I if f is a Maass form of weight 0 or
1), or the height l8'msl of the complex variable s. Today the analytic theory of
£-functions is sufficiently well developed so that the following statement can be
proved:
Subconvexity Theorem for GL 1 , GL 2 • There exists an explicit constant o > 0 such
that for any primitive Dirichlet character x, any f and g primitive modular forms,
and any s such that ~es= 1/2, one has:
- s-aspect: as Isl --+ +oo,
L(x,s) « lsll/4-o, L(f,s) « (lsl2)1/4-o,L(f0 g, s) « (lsl4)1/4-o,
the implied constants depending on qx, Q1 and Q1Q 9 , respectively;- q-aspect: as qx, qi --+ +oo,
L(x, s) « q~/4-o, L(f,s) « (qi)1/4-o,L(f@g,s) « (qJ)l/4-o,
the implied constants depending on Isl, Clsl,μ1,1) and Clsl,μ1,1), respec-
tively;- Spectral aspect: as μ1,1 --+ +oo
L(f,s) « (lμ1,1l2)1/4-8,L(f0 g,s) « (1μ1,114)1/4-8,
the implied constants depending on (Isl, qi) and (Isl, qlq 9 , μ1,1), respec-
tively.This statement (of which a few cases remain to be proved), has a long his-
tory and many contributors. The first subconvexity result is due to Weyl [We] for
Riemann's (:
(4.1)
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