Computing the Hilbert class field of real quadratic fields

Abstract

Abstract. Using the units appearing in Stark's conjectures on the values of L-functions at s = 0, we give a complete algorithm for computing an explicit generator of the Hilbert class field of a real quadratic field.Let k be a real quadratic field of discriminant, and let ω denote an algebraic integer such that the ring of integers of k is O k := Z + ωZ. An important invariant of k is its class group Cl k , which is, by class field theory, associated to an Abelian extension of k, the so-called Hilbert class field, denoted by H k . This field is characterized as the maximal Abelian extension of k which is unramified at all (finite and infinite) places. Its Galois group is isomorphic to the class group Cl k ; hence the degree [H k : k] is the class number h k .There now exist very satisfactory algorithms to compute the discriminant, the ring of integers and the class group of a number field, and especially of a quadratic field (see [3] and [16]). For the computation of the Hilbert class field, however, there exists an efficient version only for complex quadratic fields, using complex multiplication (see [18]), and a general method for all number fields, using Kummer theory, which is not really satisfactory except when the ground field contains enough roots of unity (see [6], [9] or [15]).In this paper, we will explore a third way, available for totally real fields, which uses the units appearing in Stark's conjectures [21], the so-called Stark units, to provide an efficient algorithm to compute the Hilbert class field of a real quadratic field. This method relies on the truth of Stark's conjecture (which is not yet proved!), but still we can prove independently of the conjecture that the field obtained is indeed the Hilbert class field and thus forget about the fact that we had to use this conjecture in the first place.Of course, the possibility of using Stark units for computing Hilbert or ray class fields was known from the beginning, and was one of the motivations for Stark's conjectures. Stark himself gave many examples. It seems, however, that a complete algorithm has not appeared in the literature, and it is the purpose of this paper to give one for the case of real quadratic fields.In Section 1, we say a few words about how to construct the Hilbert class field of k when the class number is equal to 2. Here, two methods can be used which are very efficient in this case: Kummer theory and genus field theory. In Section 2 we give a special form of Stark's conjectures, namely the Abelian rank one conjecture applied to a particular construction. Section 3 is devoted to the description of the

Real Multiplication Revisited

“…Along with the method of Stark's units [19], theorem 1 can be used in the computational number theory. For the sake of clarity, we shall consider the simplest examples; the rest can be found in Table 1.…”

Index formulae for Stark units and their solutions

“…8]. 3 Unfortunately, in most cases the values are complex and there does not appear to be any obvious way to remove these absolute values. 4 For v a finite place, the abelian rank one Stark conjecture is basically equivalent to the Brumer-Stark conjecture, see [15,§IV.6].…”

Topics in computational algebraic number theory

“…The Stark units algorithm of Cohen-Roblot [13] produces a relative polynomial of comparable size (the norm of its discriminant has 2485 decimal digits), but is more cumbersome: it requires about 45 minutes computational time, using 600 MBytes RAM in the Pari implementation (one can dispense with precomputations and reduce memory usage to our default 10 MBytes, roughly tripling running times). Remark 7.11.…”