1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 3. LARGE SIEVE INEQUALITIES 233

either to a dihedral group D2n, n ;:?: 1, or to one of the exotic groups A 4 , S 4 , A 5.

Accordingly, one says that f is of dihedral, tetrahedral, octahedral or icosahedral

type, respectively.

The forms of dihedral type are well understood: up to some twist, they are

given by theta series associated to (non-real) characters of the ideal class group of

imaginary quadratic orders. In particular one has the upper bound

(3.13) ISfihedral(q, x)I «,, ql /2 log q;

while Siegel's theorem and the known estimates for the size of the non-maximal

orders of imaginary quadratic fields (see for example [Cor]) give, for xis quadratic,

the lower bound

ISfihedral(q, x)I »,, ql/2-e:

for any c > 0.

The forms of exotic type are much more mysterious. In [Serl], Serre studied

in depth the possible Hecke eigenvalues of exotic forms (for quadratic nebentypus)

and found that they belong to an explicit finite set; this lead him to conjecture that

exotics forms are very rare: more precisely, that

1sfxotic(q, x)I «,, q",

for any c > 0. Duke made the first significant step in the direction of this conjecture

by improving the trivial bound by a factor q^1111 for a quadratic nebentypus x; a little

later, S. Wong extended Duke's argument to nebentypus x [Wo]:

Theorem 3.8. One has

for any c > 0.

Proof. Duke's method uses two conflicting properties of the Hecke eigenvalues of

exotic forms.

(1) Rigidity: for p ;fq, the >.. 1 (p) are frobenius eigenvalues associated to a 2-

dimensional complex galois representation p 1 ; the study of the possible

lifting of the exotic subgroups to G L 2 ( C) provides a lot of information on

the character table of Pt hence on the >..1(P) ([Serl, Wo]). In particular,

one can the following linear relation, valid for any f E Sfxotic(q, X) and

any prime p ;{'q,

(3.14) x^6 (p)>..1(P^12 ) - x4(p)>..1(P^8 ) - x(p)>..1(P^2 ) = l.

It implies that the >.. 1 (p) are algebraic integers that can take only finitely

many values (depending on the order of x ).

(2) Orthogonality: the quasiorthogonality relations among eigenvalues are

encapsulated in (3.10) and reflect their modular nature.

One chooses an as follows:

{

x^6 (p), for n = p^12 ~ N, (p, q) = 1

an= -x4(p), for n = ps ~ Ns/12' (p, q) = 1

-x(p), for n = P2 ~ N2/12' (p, q) = 1

0 else.