50 L. CLOZEL, SPECTRAL THEORY OF AUTOMORPHIC FORMS
The theorem was proved by Eichler and Shumura for k = 2, then by Deligne
for k ;=:: 2, and then "descended" to k = 1 by Deligne and Serre. See [54, 20, 21].
Since this is indispensible to understand the origin of the higher-dimensional cases
we sketch the proof for k = 2. Assumer= r 0 (N). (We also assume that w = 1).
Then Y 0 (N) = f\'H is a smooth complex affine curve. It has a compactification
X = X 0 (N) (add the cusps). One knows the following facts:
(1) X is naturally defined over Q.
(2) It in fact has a natural model over Z[*], and therefore over IFP for pf N.
This reduction mod pis smooth over IFP.
We now choose p and think of X as a curve over IFP. On X we have the
following correspondences (a correspondence here is just a curve C C X x X,
with no horizontal or vertical components).
(a) The Frobenius morphism 'Pp·
We think of this in the old style: embed X into IP'N, and consider the map
X 3 x = (xi) 1--> (xf). If X c IP'N is defined over IFp, this gives a IFp-rational
morphism X ---) X; C is its graph.
(b) The Hecke operator Tp.
Tp can be seen as an algebraic correspondence on the complex curve X; with
a suitable definition it makes sense over !Fv.
( c) The transposed Frobenius t'PwHere transposition just means that we replace CCX x X by tc, obtained by
exchanging the factors of X x X.
Now the basic theorem (Kronecker, Weber, Eichler, Shimura) is
(1.9)as correspondences on X - there is a notion of sum of correspondences. (See
Shimura [ 54 ,Ch. 7] and Labesse [38]).
Consider the HI of X. Over C, we have S 2 (f\'H) = DI(X(C)) (space of
holomorphic 1-forms) and therefore (Riemann)
(1.10)
where S 2 is the space of anti-holomorphic forms of weight 2. This is the trivial case
of the Eichler-Shimura isomorphism [54, Ch. 8].
Now correspondences act on cohomology spaces. Over C we take Betti coho-
mology on X(C) as in (1.10). Over Fv we take etale cohomology with coefficients
in Qe. By standard comparison theorems,
This is compatible with the action of Tp; in order to control the eigenvalues in S 2 ,
it suffices by (1.10) to control the eigenvalues in HJt(Xwv,Qe). Now we use (1.9).
When cpp acts on HI, we have the further relation
(1.11)