1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 5. SOME APPLICATIONS OF SUBCONVEXITY

where W ( 'ljJ, V) denotes the "twisted" Weyl sum

1

W('l/;, V) = IPic(OK)I L ~(cr)V(Eg).

aEP1c(OK)

For V(z) = E 00 (z, 1/2 +it) Hecke has shown (see [DFI4] for a proof) that

. WJ{ L('l/;, 1 /2 + it)IDl^1 /4+it/2
W('l/;, Eoo(z, 1/2 +it)) = 23/2+it ((1 + 2it) IPic( OK )I

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In particular we deduce from Siegel's theorem, Burgess's bound (when 'ljJ is real),

and Theorem 4.1 (when 'ljJ is complex) that

W( G, E 00 (z, 1/2 +it)) = Ot( IPi~~K) I D - 1/24000.5).

When V(z) = f(z) E Sf{(l, t) is a primitive weight zero Maass cusp form and

'ljJ = 'I/Jo is the trivial character, Maass has shown the formula

vf2n-1/41Dl3/4

W('l/Jo,J) = IPic(OK)I Pj(-D),

where Pj(-D) denotes the Fourier coefficient of a Maass form of weight 1 /2 and

eigenvalue 1/4 + (t/2)^2 corresponding to f by a theta-correspondence (see [Du]).

For such forms J, Duke proved directly that (5.6) holds, from which it follows that

W('l/Jo, !) = o1,,:(D-1/2s+"),

which proves (5. 9) for the full orbit ( G = Pie( 0 K); eventually one could also have

used Waldspurger's formula to relate non-trivial estimates for Fourier coefficients

of half-integral weight Maass forms to the ScP.

The method of Maass, however, does not seem to generalize to the twisted

Weyl sums W ( 'ljJ, f) when 'ljJ is not a real character. Recently, Zhang, in a wide

generalization of the Gross/Zagier formulas, related twisted Weyl directly sums to

central values of Rankin/Selberg £-functions [Zl, Z2, Z3]: for any character 7/J of

Pic(OK), one has

(5 .11) IW(·'· !)12 = (f ·'·) D1/2 L(f © B,μ, 1/2)

y;, c ,y; IPic(OK)l^2 '

for some constant c(f, 'ljJ) « 1 that is uniformly bounded in 'l/;; the implied constant

depending on f only. Hence any subconvexity exponent for L(f ©B,μ, 1/2) in the D-
aspect is sufficient to show (ineffectively) that W('l/;, !) = o 1 (D-^6 ) for some o > O;
in particular, o = 1/2200 holds, and this concludes the proof of (5.9). Note that the
limitation on the index of G comes from the Eisenstein spectrum.
The proof of (5.10) is very similar to (5.9), once the problem has been formu-
lated in the appropriate context (for this we refer to [Gr, BDl] for more details).
We denote by M = @eau•• (F •) Z e the set of divisors supported on Eef^88 (F q) and

by M^0 the kernel of M under the degree map:

deg(L ne.e) = L ne.

e e

M is equipped with a natural inner product given by