1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. ANALYTIC PROPERTIES OF INDMDUAL L -FUNCTIONS 199

Remark 1. 7. Although Siegel's Theorem does not strictly eliminate the exceptional

zero, it is rather sharp. (For instance, it implies that the asymptotic (1.17) holds

uniformly for q in the range q «A (log x) A for any A ;;;:: 1, the implied constant

depending only on A.) However, the above constant c(c) depends also on the

hypothetical Xex, and in particular, cannot be computed effectively (for any c <

1 /2). This is the major drawback of the Theorem, since, as we shall see below, the

question of the effectivity can be a very important issue.

Remark 1.8. Later, Tatuzawa gave a slightly different formulation of this result
[Ta] : for any c > 0, there exists an effectively computable constant c( c) > 0 such that
for all quadratic characters x, with but at most one exception, L(x, s) has no zero in
the interval
[1 - c(c)/q~, l].

1.2.2.1. Dirichlet Class Number Formula and the Class Number Problem. Dirichlet

gave several proofs of the non-vanishing of L(x, 1) for quadratic characters; one is

a direct consequence of his Class Number Formula. Given x ( mod q), a primitive

odd (resp. even) quadratic character, one has

2nhK logcKhK

(1.20) L(x, 1) = 112 , (resp. = 112 )

WJ<q q

where K = Q(Jx(-l)q) denotes the associated quadratic field, WK the order of

the group of units in K (if K is imaginary), cJ< > 1 the fundamental unit (if K is

real) and hK := JPic(OK)I denotes the Class Number. In particular, since hK ;;;:: 1

(and cJ< ;;;:: vq/2 if K is real), one has L(x, 1) » q-^1 /^2 ; this and (1.19) imply

(1.18).

A famous conjecture of Gauss (formulated in the language of binary integral
quadratic forms) is that there are finitely many imaginary quadratic fields with
given class number; assuming this conjecture and given some h ;;;:: 1, the Class
Number Problem asks for the list of all imaginary quadratic fields K with class num-
ber JPic(OK)I = h. Thus the Class Number Formula provides a way to approach
Gauss's conjecture by analytic methods.
Eventually, Gauss's conjecture was solved by Heilbronn by proving a weak form
of Siegel's Theorem; for instance, if follows from (1.19) and (1.20) that, for any
c > 0,
[Pic(OK)[ »e: q^1 /^2 -"';
in particular [Pie( OK)[~ +oo as q =[Disc( OK)[~ +oo.
However, because of the lack of effectivity in Siegel's theorem, this does not
solve the Class Number Problem; at best, it follows from Tatuzawa's version dis-
cussed above that in principle (i.e. with a sufficient amount of computer assis-
tance), a list of all imaginary quadratic fields with Class Number h can be given, up
to possibly one missing field.
For h = 1 and 2, the Class Number Problem was solved independently and at
the same time by Baker and Stark (in 1966 for h = 1 and in 1970 for h = 2),
each one using quite different methods. In fact, it was recognized later (by Birch
and Stark) that an earlier solution (going back to 1952) by Heegner of the Class
Number One Problem was correct. For general h, the Class Number Problem has
been solved, in principle (i.e. with a sufficient amount of computer assistance),
by the conjunction of the works of Gross/ Zagier and Goldfeld [Go2, GZ]. Their