Lecture 2. A review of classical automorphic forms
In this section we review the theory of GL 2 ,q-automorphic forms from the
classical point of view, following Maass and Selberg, and collect various estimates
and technical formulas that will be used later. Some of this material is borrowed
from the exposition given in Sections 4/ 5/ 6 of [DFI8].
2.1. Spaces of Holomorphic and Maass forms
The group SL 2 (R) acts on the upper half-plane by fractional linear transformations
az + b. ( a b )
/Z = CZ+ d , if I = c d.For/ E SL2(R), we define the multipliers
j(r, z ) = (cz + d),
CZ +d
j 1 (z) = lcz +di = exp(i arg(cz + d)),and for any integer k ~ 0, two actions (of weight k) on the space of functions
f : H ----> C, given by
J 1 k 1 (z ) = (cz + d)-k f(rz ).
k , (z ) = j 1 (z)-kf(rz).
For q ~ 1, we denote by r the congruence subgroup r 0 (q) = { ( ~ ~ ) E
SL 2 (Z), c = O(q)}; then any character x(mod q) defines a character of r by the
formula
2.1.1. Holomorphic forms
For k ~ 1, we denote Sk(q, x) the space of holomorphic cusp forms of weight k,
level q, and nebentypus x (i.e. the space of holomorphic functions F : H ----> C that
satisfy
(2.1)
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