LECTURE 5. SOME APPLICATIONS OF SUBCONVEXITY 273and in particular by Deligne's bound for holomorphic cusp forms Cl.A 9 (n)I ~ T(n)),
it is sufficient to prove (5.6) for n = d squarefree. In the latter case, one can
proceed directly as in [Il] or use Waldspurger's formula, which relates the Fourier
coefficient pf ( d) to a central value of a twisted L-function. If D is the discriminant
of some quadratic field K, with n^1 -^1 /^2 > 0, then Waldspurger's formula has the
form
IP1(IDl)l^2 = C(f, g, D)L(g.xK, 1/2),
where XK is the associated Kronecker symbol and C(f, g, D), the proportionality
constant, is bounded independently of D. In particular, for
D = Dise(Q( V(-l)l-1/2d))
we see from (5.8) (eventually after a Mobius inversion) and Deligne's bound, that
d^112 P1(d) <<t:,J de:IL(g.xK, 1/2)1^1 /^2.Hence, any improvement over the 1/4 exponent in (5.5) is equivalent to the so-
lution of the ScP for L(g.xK, 1 /2) at the special point s = 1 /2 in the D aspect,
and GLH implies the analog of the Ramanujan/ Petersson Conjecture. In particular,
Theorem 4.8 yields
P f (d) « e:,f d-l/4-l/44+e:
which is slightly weaker than (5.6). On the other hand, (5.6) (whose proof doesn't
make use of L-functions) yields, for any primitive form g, the subconvex bound
L(g.xK, 1/2) «e:,g IDl1/2-1/14+e:.Remark 5.5. Recently Baruch and Mao [BM] have generalized Waldspurger's for-
mula to modular forms over totally real fields. This is being used in [CoPSS] in the
resolution of the last cases of Hilbert's 10-th problem.
5.3.2. Equidistribution of Heegner points
Given K an imaginary quadratic field, we denote by Ef£of( (Q) the set of (iso-
morphism classes of) elliptic curves over Q with complex multiplication by 0 K.
By the theory of complex multiplication, these curves are defined over HK, the
Hilbert class field of 0 K, and Gal (HK/ K) c::: Pie( 0 K) acts simply transitively on
EUo I< ( Q); for O" E Pie( 0 K), we denote by E^17 the corresponding Galois action on
E.
To any given place q of Q over some place q of Q, we associate a map r q as
follows:
- If q = oo is the infinite place, abusing notation, we denote by oo the
corresponding embedding q : Q--) C; for EE EU 0 I< (Q), the associated
complex curve is the quotient of C by a lattice, E(C) = C/(Z + zEZ),
where ZE E H is defined modulo f 0 (1). We set
r. Ef£of((Q) --) EU(C) = X 0 (1)(C) - {oo}(^00) • E --) ZE (mod fo(l)).
- If q is above some finite prime q, we define r q(E) := redq(E) to be the
reduction modulo q of E. If q splits in K, then E reduces to an elliptic
curve over F q with complex multiplication by OK, and rq sends E££01< (Q)
bijectively to EU 0 I< (F q), the corresponding set of (isomorphism classes
of) elliptic curves over F q; in that case there is not much for us to say.