274 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS- On the other hand, if q is inert in K, then the reduction map restricted to
£££ 0 /( (Q) is not injective in general; indeed, E E £££ 0 /( (Q) reduces to
a supersingular elliptic curve (with complex multiplication by some max-
imal order in the definite quaternion algebra ramified at q). We denote
by Enss (F q) = { e 1 , ... en} the (finite) set of (isomorphism classes of)
supersingular elliptic curves in characteristic q (in fact, these curves are
defined over Fq2); we have the map
. ERfoI<(Q) , Effss(Fq)
rq · E , rq(E).
The spaces Xo(l) (C)-{ oo} and Enss (F q) each carry a natural probability measure.
For the former this is simply the Poincare measure μ 00 (z) = (3/w)dxdy/y^2 , which is
induced by the hyperbolic metric. The measure μq on Effss (F q) is a bit less obvious
(however, we refer to the lecture of Gross [Gr] and to the first section of [BDl] for
an explanation of why this is indeed, the natural measure on this space); for any
e E xgs (1)(Fq2), we set( )
l /we
μq e = -"""""'---,
L 1/we'
e'E£f£^88 (Fq)
where We = I End( e) x I is the number of units of the (quaternionic) endomorphism
ring of ei. Note that μq is not exactly uniform, but almost (at least when q is large),
since 1£££ss(Fq)I = n = 9fj-+ 0(1) and the product w 1 ... wn divides 1 2.
As we will see, it follows from certain cases of the ScP that for each q, the
images r q ( ERfo K ( Q)) are equidistributed on the corresponding spaces relative to
the corresponding measure μq, as IDisc( Ox) I _, +oo. In fact one has the following
stronger equidistribution result:Theorem 5.4. For each K, pick E 0 E £££oK (Q) and G c Pie( Ox), a subgroup of
index i ~ I Disc( Ox )1^1 /^24001. Then for any continuous V on Xo(l)(C) one has, as
IDisc(Ox )I _, +oo,(5 .9) W(G, V) := l~I L V(roo(Eg)) = r V(z )dμoo(z ) + ov(l).
uEG j Xo(l)
For any place q above some finite prime q that is inert in K, and for any function V
on Euss (F q) one has(5.10) l~I L V(rq(E 0 )) = L V(e)μq(e) + ov,q(l).
uEG eEt:fiss(Fq)Proof. By Weyl's equidistribution criterion, and in view of the spectral decomposi-
tion (2.5) for X 0 (1)(C), it is sufficient to prove (5.9) when V(z) is either a Maass
cusp form (which we may also assume to be a Hecke form), or the Eisenstein se-
ries Eoo(z, 1/ 2 +it); in either case, one has fxo(l) V(z)dμ 00 (z ) = 0. By Fourier
transform, one has
W(G, V) = l~I L V(r 00 (E 0 )) = L W(7j;, V),
uEG ~EPic{Q;<)
~a=l