1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. MOSTLY SL(2) 49

In all cases we see that p-^1 /^2 < lapl, l,8pl < p^112 ("Hecke's trivial estimate"). Now

we can state:

RAMANUJAN CONJECTURE (for Maass forms).

iapl = l,Bpl = 1 , or, equivalently: lapl ::::; 2,/P.

Finally, we note t hat the general case of Maass forms for congruence groups is

in fact a little more complicated. One must consider functions invariant by

f1(N)={I= (~ ~) Ef:c:=O[N],a:=d:=l[N]}

and transforming under f 1 (N)\f 0 (N) ~ (Z/Nz)x by a character w. There are

again natural operators Tp which a re normal and have all other properties. The

previous computations apply, with now

ap,Bp = w(p) (pf N).

1.4. We now review the sit u ation for classical modular forms.

Fix k :'.'.'. 2. R ecall that Mk(r), t he space of (classical) modular forms of weight

k, is given by

Mk(r) = {f holomorphic on H , f G:: ~) = (cz + d)kf(z)}.

Then Sk(r) c Mk(r) is the space of cusp forms. There is no Laplacian (replaced by
the condition 8 f = 0 ... ) but there are, again, Hecke operators given by formulas
similar to (1.3) - see Shimura [5 4 ].
There is a natural normalization of t he Tp such that
(1.6) Tpf = apf , f E Sk(r) ===} ap is an algebraic integer
and minimal for this property (see [54] et [41]). We adopt t his normalization. Then,
for varying p (f a fixed eigenform)

(1.7) lapl = O(pkf^2 ) (Hecke's trivial estimate).

For forms on r 0 (N) (possibly with a character w) these definitions apply for
pf N. We then have

RAMANUJAN CONJECTURE (holomorphic case). For a given eigenform
f , eigenvalues (ap),

As for Maass forms we can use the "unitary normalization". We write ap =

p k2

1
(ap + ,Bp) ; ap,Bp = w(p) and then the conjecture is equivalent to
(1.8)
Finally, t h ere are forms of weight k = 1, and the Conjecture applies to t hem,
giving

Theorem 1.1. - Assume f E Sk(r 0 (N)), k > 1 is an eigenform. Then the
Ramanujan conjecture is true for f.