Extreme values of Artin L-functions and class numbers

Assuming the GRH and Artin conjecture for Artin L-functions, we prove that there exists a totally real number field of any fixed degree (> 1) with an arbitrarily large discriminant whose normal closure has the full symmetric group as Galois group and whose class number is essentially as large as possible. One ingredient is an unconditional construction of totally real fields with small regulators. Another is the existence of Artin L-functions with large special values. Assuming the GRH and Artin conjecture it is shown that there exist an Artin L-functions with arbitrarily large conductor whose value at s = 1 is extremal and whose associated Galois representation has a fixed image, which is an arbitrary non-trivial finite irreducible subgroup of GL(n, C) with property GalT . 1 Class numbers of number fields. Let K be a number field whose group of ideal classes has size h, called the class number of K. As K ranges over some natural family, it is interesting to investigate the behavior of h. Unless K is imaginary quadratic, the involvement of an infinite unit group makes certain problems here extremely difficult, even assuming conjectures like the generalized Riemann hypothesis (GRH). An obvious example is Gauss’ conjecture that there are infinitely many real quadratic fields with h = 1. On the other hand, some rather precise information can be obtained about number fields with large class number. Consider, for example, the family Kn where K ∈ Kn if K is a totally real number field of degree n whose normal closure has the full symmetric group Sn as its Galois group. By the class number formula for such K h = d 2n−1R L(1, χ) (1) where d = disc(K) is the discriminant, R is the regulator and L(s, χ) = ζK(s)/ζ(s) is an Artin L-function, ζK(s) being the Dedekind zeta function of ∗Research supported by NSF grant DMS-98-01642.