Computing the lead term of an abelian L-function

Dummit, David S.; Tangedal, Brett A.

doi:10.1007/bfb0054879

Cited by 8 publications

(10 citation statements)

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“…Since then, hundreds of new examples have been computed over a variety of different base fields in [St4,DST1,DT,DTvW], always giving full confirmation of the refined conjecture to within the accuracy of the computations. Stark's conjecture may be used as an effective algorithm for computing an explicit generating polynomial of the Hilbert class field of a real quadratic field (see [St1] and [CR]).…”

Section: Stark's Refined Conjecture Over a Real Quadratic Fieldmentioning

confidence: 92%

Continued fractions, special values of the double sine function, and Stark units over real quadratic fields

Tangedal

2007

Journal of Number Theory

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Let F be a real quadratic field and m an integral ideal of F. Two Stark units, ε m,1 and ε m,2 , are conjectured to exist corresponding to the two different embeddings of F into R. We define new ray class invariants U (1) m (C + ) and U (2) m (C + ) associated to each class C + of the narrow ray class group modulo m and dependent separately on the two different embeddings of F into R. These invariants are defined as a product of special values of the double sine function in a compact and canonical form using a continued fraction approach due to Zagier and Hayes. We prove that both Stark units ε m,1 and ε m,2 , assuming they exist, can be expressed simultaneously and symmetrically in terms of U (1) m (C + ) and U (2) m (C + ), thus giving a canonical expression for every existent Stark unit over F as a product of double sine function values. We prove that Stark units do exist as predicted in certain special cases.

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Section: Stark's Refined Conjecture Over a Real Quadratic Fieldmentioning

confidence: 92%

Continued fractions, special values of the double sine function, and Stark units over real quadratic fields

Tangedal

2007

Journal of Number Theory

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“…For more details, see Lemma 5.1 of [12].) The values of L ð2Þ ð0; * wÞ can then be computed using the method of [2].…”

Section: Remarks On Computational Methodsmentioning

confidence: 99%

“…Section 3 develops a new formula to compute the element Φ f,T,p (1). We concentrate on the case where k is real quadratic although our technique should extend to other totally real fields.…”

Section: Introductionmentioning

confidence: 99%

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Verifying a p-adic abelian Stark conjecture at s=1

Roblot

Solomon

2004

Journal of Number Theory

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“…We will essentially follow the method given in [8] with a slightly different computational approach. This method needs to work with the conductor of the character χ, but thanks to lemma 3.2, we know that this conductor is f = f 0 v for odd characters.…”

Section: Algorithm 35 Let a > 0 Be A Real Number And Let N ≥ 1 Be Amentioning

confidence: 99%

Computing the Hilbert class field of real quadratic fields

Cohen

Roblot

1999

Math. Comp.

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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

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Computing the lead term of an abelian L-function

Cited by 8 publications

References 6 publications

Continued fractions, special values of the double sine function, and Stark units over real quadratic fields

Continued fractions, special values of the double sine function, and Stark units over real quadratic fields

Verifying a p-adic abelian Stark conjecture at s=1

Computing the Hilbert class field of real quadratic fields

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