Extreme Values of Artin L-Functions and Class Numbers

Abstract

Assuming the generalized Riemann hypothesis (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 nontrivial finite irreducible subgroup of GL(n, ℂ) with property Gal T .