1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 5. SOME APPLICATIONS OF SUBCONVEXITY 269

5.2. Subconvexity vs. Minkowski

Let K be an imaginary quadratic field. We denote by -D = Disc( 0 K) its discrim-

inant, Pie( 0 K) the ideal class group of 0 K, 'ljJ a character of Pie( 0 K), and 'I/Jo its

trivial character.

Question. Given 'ljJ a non-trivial character of Pie( 0 K ), what is the smallest N ( 'ljJ)

such that there is an ideal a c OK of norm NK;q(a) = N('l/J) satisfying 'ljJ(a) =f=. l?

This question is very similar to the previous one. By Minkowski's theorem each

ideal class of Pic(OK) is represented by an integral ideal of norm::::;; (2/7r)VD,

hence

N('ljJ)::::;; (2/7r)vlf5.

On the other hand, we can proceed as before and evaluate the partial sum

°'£v('l/J, N) := L 'ljJ(a)V( N K~(a)) =

2

:i J L('ljJ, s)V(s)N^8 ds;

ac01< (3)

here

It is well known (from Hecke), that L( 'ljJ, s) is the £-function of the theta series

B,p(z ) := L e2itrNI<;Q(a)z E M1(D, XK),

aCOg

which is a holomorphic modular form of weight one and nebentypus XK, the qua-

dratic Dirichlet character associated with K. Moreover, B,p(z) is primitive, and it is

cuspidal unless 'ljJ is real, in which case B,p(z ) is an Eisenstein series and (Kronecker's

formula) L('ljJ, s) factors as a product

L('l/J, s) = L(x1,,p, s)L(x2,,p, s)

corresponding to a (uniquely determined) factorization of the Kronecker symbol

XK into a product of two primitive quadratic characters XK = x 1 ,,px 2 ,,p. In partic-

ular, for the trivial character one has L('ljJ 0 , s) = ((s)L(xK, s), hence, shifting the

contour to ~es= 1/ 2 we hit a pole at s = 1 only when 'ljJ = 'ljJ 0. Hence we have

°'£v('l/J, N) = Dw=..Po V(l)L(XK, l)N +

2

:i J L('ljJ, s)V(s)N^5 ds

(1/ 2)

(S.2) Dw=..Po V (l)L(XK, 1 )N + O(N1;2 Dl/4-1/ 24 000)

by the subconvexity bounds (either Burgess's bound if 'ljJ is real or Duke/ Fried-
lander/Iwaniec's bound of Theorem 4.1 otherwise). Taking 'ljJ =f=. 'I/Jo and N ::::;; N ( 'ljJ)
we obtain by Siegel's theorem (1.19), and the identity °'£v ( 'ljJ, N) = °'£v ('I/Jo, N), the
bound
N('l/J)::::;; n1/2-1; 12001;
here the implied constant is absolute but ineffective. Another kind of application
that can be obtained along these lines is the following: suppose that Pie( OK) con-
tains a cyclic subgroup G of small index. (For example, the Cohen-Lenstra heuris-
tics predict that there are infinitely many prime discriminants such that Pie( 0 K)