1549380232-Automorphic_Forms_and_Applications__Sarnak_

216 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS

S'k(q, x, it)) is determined up to scalars by all but finitely many Hecke eigenvalues;

as a consequence a new Hecke eigenform is automatically an eigenform of all the

Tn· We denote by S f (q, x, it) the set of primitive forms of S'k(q, x, it): i.e. the set

of eigenforms for the full Hecke algebra normalized by PJ(l) = 1. Primitive forms
form orthogonal basis of S'k(q, x, it) and by the strong multiplicity one property
primitive forms in S'k(q, x, it) are also eigenforms of the operator Qit ,k> hence one
has (PJ(-1) =OJ PJ(l)). Primitive forms are also "quasi-eigenvalues" of the Atkin-
Lehner-Li operators W qp which are defined, for each q 1 lq such that (q1,q/ q1) = 1,
by the I k action of some matrices Wq 1 of determinant q 1 , which normalize r 0 ( q). We

have Wq 1 : Sk(q, x, it) f--f S k(q, Xq 1 Xq/qi, it) where x = Xq 1 Xq/qi is the factorization

of x into characters of moduli q 1 and q/ q 1. Then for f a primitive form, one has

W qJ = w J (q 1 )g, where g is primitive and wJ(q 1 ) has modulus one (see [AL, ALi]).

2.3. Classical L-functions vs. Automorphic L-functions

Given f E S f (q, x, it) a primitive form, Hecke associated to it a classical £-function

and proved its basic analytic properties (see [15]). There are also other £-functions

related to classical modular forms, namely the £-functions L(f x g, s) associated to

pairs (!, g) which were initially studied by Rankin and Selberg, and the symmetric

square £-function L(sym^2 f , s ), which was first defined and studied by Shimura

([Sh]). These are Euler product of degree 4 and 3 respectively and one of their

main advantages is that the coefficients of these Dirichlet series are easily expressed

in terms of the Hecke eigenvalues off (and g), as one can see from the expressions

below. However, the major drawback of considering these £-functions from the

classical viewpoint is their functional equation, which may be quite complicated

and hard to prove in full generality, especially when levels are divisible by high

powers; to be convinced, the reader may want to have a look at the paper of W.

Li [Li2], which derives the functional equation for Rankin/ Selberg £-functions by

classical means.

On the other hand, a well known recipe associates to f an automorphic repre-

sentation, 7r J (say), with the same conductor [Ge]. Its local parameters are related

to the Hecke or Laplace eigenvalues as follows:

AJ(P) = 0:7r 1 ,1 (p) + 0:7r 1 ,2(p), 0:7r 1 ,1 (p )o:7r 1 ,2(P) = X(P)

at the non-archimedean places ( in particular >.7r 1 ( n ) = >. J ( n )) ; for the archimedean

places one has the identities

1-OJ.

-μJ,l = -2-+it

1-OJ.

-μJ,l = -2-+ it

k-l

-μJ,l = - 2-

l- 01.

'-μJ,2 = - 2 - - it,

'-μJ,2 = - it,

'-μJ,2 -- - 2-k+l

l8'mt1< 1/ 2

t ER

depending whether f is a Maass form of weight k = 0 or k = 1 (here o J is the

eigenvalue for the operator Qit ,k), or is holomorphic of weight k ;;::: 1, respectively.