270 PH. MICHEL, ANALYrIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS
is cyclic.) One may then look for an ideal a generating G of small norm; from
Minkowski's bound there is such a generator with norm bounded by
NK;q(a) ~ (2/7r)VD.Theorem 4.1 allows us to improve on the exponent 1/2:
Theorem 5.1. Let G c Pie( 0 K) be a cyclic subgroup of index ic. Then G is generated
by an ideal of norm « ibD^112 -^1 /^24001 , where the constant implied is absolute but
ineffective.
Proof. We denote by 8c(.) the characteristic function of the generators of G; to
show that there is a generator a c 0 K of norm ~ N, it is sufficient to show that
L-v(G,N) := L 8c(a)V(NK~(a)) # 0.
aC01<Let g c G, be a generator of G by easy Fourier analysis and Mobius inversion, one
has
1
Sc(a) = IPic(OK)I 0._ 'lf;(a)
,PEPic(O I<)L 1j}(gm) = ;__ L μ~) L 'if;(a).
m(JG J) ic dJJGI .Pfa=l
(m,JGJ)=lFrom (5.2) it follows that
L-v(G, N) = ~ L μ(dd) L l'-v('i/J, N)
ic dJIGI ,μd
1 a=l= i~ cp~~I) V(l)L(xK, l)N + O(T(IGl)N1;2 n1/4-1/24ooo),
so by Siegel's theorem this sum is nonzero when N » ibD^114 -^1 /^24001. O
Remark 5.3. In both examples, the convexity bound essentially matches Minkow-
ski's. Note also that Minkowski theorem is also based on a convexity argument
(although quite different from the Phragmen/ LindelOf principle); moreover, via
Arakelov geometry, Minkowski's theorem for number fields can also be seen as an
analog of the Riemann/ Roch Theorem for curves; for more on this see Szpiro's
"Marabout Flash" [Szp].
5.3. Subconvexity and distribution of Heegner points
In this last section we describe applications of the subconvexity bound to various
equidistribution problems.
5.3.1. Equidistribution of lattice points of the sphere.
Given n ;:;:: 1, it goes back to Gauss that n is representable by the ternary quadratic
form X^2 + Y^2 + Z^2 if and only if n is not of the form 4k(8l - 1). We denote by
R 3 (n) := {:X = (x, y, z) E Z^3 , x^2 + y^2 + z^2 = n} (resp. R3(n) := {:X = (x, y, z) E
Z^3 , x^2 + y^2 + z^2 = n, g.c.d(x,y,z) = 1}) the set of representations of n as the
sum of three squares (resp. of the primitive representations) and by r 3 (n) (resp.
rj(n)) the number of such representations. We have r 3 (n) = I:d 2 ln r3(n/d^2 ); on
the other hand, Gauss gave a formula for the number of primitive representations