1549380232-Automorphic_Forms_and_Applications__Sarnak_

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

is in fact a holomorphic modular form of weight l = k + 3/2 and level 4 and a cusp

form when k? 1 (see [15] Chap. 10). For a cusp form f of half-integral weight l

of some level q = 0(4) with Fourier expansion

f(z) = L P1(n)n~ e(nz)

n;;. 1

one has the following bound of Hecke type which comes from Petersson's formula

(either for holomorphic forms or for half-integral weight Maass forms) (see (15]

chap. 5):

(5.5) P f (n) « f ,c n-l/4+c

for any c > 0. This bounds yields r p ( n) «P,c n kt

1

+^0 which is barely not suf-

ficient for our equidistribution problem. In fact, either for squarefree n or for

weight l? 5/2, Hecke's bound is not sharp: one expects the analog of the Ra-

manujan/ Petersson Conjecture

P f (n) « f,c n-l/2+c

for all n and l? 5/2, or for squarefree n when l = 3/2. We see that equidistribution

follows from any non-trivial improvement over the - 1/4 exponent in (5 .5); such an

improvement was provided for the first time by Iwaniec (11] and is given below. 0

Theorem 5.3. Let f be a holomorphic cusp form of weight l? 3/2, one has,

•for any n if l? 5/2,

• for n squarefree if l = 3/2

(5.6) P f (n) « c,f n-l/4-l/28+c

for any c > 0 the implied constant depending on c and f.

Remark 5.4. The case l? 5/2 (which is the only case of interest for the equidis-

tribution problem) was proved by Iwaniec for n squarefree (11]; the case l = 3/2

was proved by Duke by using a version due to Proskurin of Petersson's formula for

3/2-weight Maass forms [Du]; the reduction from squarefree n to all n follows from

the Shimura lifting, as is explained in [Sal] or below.

We will not give the original proof of Theorem 5.3, but rather we explain a

slightly weaker bound connected with the ScP. This is achieved through the theory

of the Shimura lift and the work of Waldspurger [Sh, Wal]. First we may assume

that f(z) is a primitive half-integral weight cusp form (in particular an eigenvalue

of the Hecke operators TP 2 defined by Shimura). Then for d squarefree, one has

(5.7)

where x(p) = C-;,1)^1 -^112 (~) and >.. 9 (p) are the Hecke eigenvalues of some primitive

holomorphic form of weight 2l - 1 and level dividing the level of f; moreover,

as long as f is orthogonal to the theta functions of one variable (for example if

l? 5/2), then g is a cusp form [Sh]. Hence for n? 1, one has

2 "'\"""""' μ(b)

(5.8) PJ(dn ) = PJ(d) L.., b 1 / 2 x(b)>.. 9 (n/b),

bin