200 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONSsolution comes from a weak (but effective) lower bound for L(XK , 1) (see [O] for
the derivation this very explicit version).
Theorem (Goldfeld/Gross/Zagier). Let K be an imaginary quadratic field with
discriminant -q. Then
1 1 1
(1.21) IPic(OK))I? - IT (1 + -)-^6 (1 + -)-^2 (1ogq).
(^55) pq I p VP
Thus with a sufficient amount of computer assistance, the CNP can be solved
for each h. This has been worked out effectively for all h ~ 16 and for all odd h ~ 23
[Ar] and recently for all h ~ 100 [Wal]. The proof of (1.21) splits into two parts:
firstly, Goldfeld [Go2] showed that if there exists a GLz-L-function vanishing with
order at least 3 at 1/2 then (1.21) holds; then, later, Gross/ Zagier [GZ] established
the existence of such an £-function (in fact the £-function of a modular elliptic
curve), as a consequence of their formula connecting the height of Heegner points
to derivatives of L-series. We refer to the survey by D. Goldfeld for an account of
the proof of this magnificient result and an update on the new cases of the Class
Number Problem that have been treated so far [1].
Recently Iwaniec/ Sarnak proposed a very interesting approach to rule out the
exceptional zero of Dirichlet characters. This approach, which unfortunately we
cannot describe here, builds in an essential way on families of automorphic £-
functions and amounts to showing that for sufficiently many primitive holomorphic
cusp forms f of some auxiliary level q', the central L-value L(f, 1/2) is large. A
very interesting point here is that the auxiliary level q' depends weakly on K: q'
may be as large as an arbitrary large power of IDisc(K)I, and, for instance, if q' is
prime, it must be inert in K. Unfortunately, this approach has not been successful
so far, although by the existing techniques one seems to be tantalizingly close to
the solution. For more information on this approach we refer to [IS3], and for
some of the most advanced ingredients that may be useful to this approach see
[DFIS, DFI6].
1.2.3. The Landau/Siegel zero for automorphic £-functions of higher degree.
For general d, the analog of the Landau/ Page theorem holds [HR]:
Theorem 1.5. Given d? 1, there is a computable constant cd such that for Q? 2,
there is at most one 1f E A~(Q) with Q1f ~ Q and L (1f, s ) having a zero in the interval
[1 - cd/ log Q, 1 [. Moreover; such 1f is self-dual and the zero is simple.
It follows, for example, from an application of Lemma 1.2.1 to
D(s) = L(II ® II, s) = ((s)L(7r1, s)^2 L(7r, s)^2 L(7r1 ®7r, s)^2 L(7r1®7r 1 , s)L(7r ® 1f, s),
and where 1f, 1f 1 are self-dual with Q7r, Q7r 1 ~ Q and II= 1EB1f EB 7r 1.
Remark 1.9. On the other hand, the analog of Siegel's theorem doesn't seem to be
known for general automorphic £-functions, but see below
It is quite remarkable, however, that in a significant number of cases the very
existence of a Landau/ Siegel zero can be ruled out.
- The first cases are due to Stark [St2], who has shown that the Artin £ -
function ( F ( s) of a Galois extension F / Q has no Landau/ Siegel zero un-
less it contains a quadratic field.