Computing Stark units for totally real cubic fields

Dummit, David S.; Sands, Jonathan W.; Tangedal, Brett A.

doi:10.1090/s0025-5718-97-00852-1

Cited by 15 publications

(6 citation statements)

References 18 publications

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“…However, if one gets results, it is possible to verify them unconditionally. Currently those methods work for small (degree ≤ 4, discriminant < 600000) totally real fields [3,8,18]. However, if these methods are applicable, they are usually quite fast, especially for large cyclic factor grops.…”

Section: Comparisonmentioning

confidence: 99%

Computing class fields via the Artin map

Fieker

2000

Math. Comp.

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Abstract. Based on an explicit representation of the Artin map for Kummer extensions, we present a method to compute arbitrary class fields. As in the proofs of the existence theorem, the problem is first reduced to the case where the field contains sufficiently many roots of unity. Using Kummer theory and an explicit version of the Artin reciprocity law we show how to compute class fields in this case. We conclude with several examples. PreliminariesLet k/Q be an algebraic number field; we denote its ring of integers by o k . A congruence module m formally consists of an integral ideal m 0 and a formal product m ∞ of real places (.) (i) of k viewed as embeddings into R. By I m we denote the set of all ideals coprime to m 0 , by P m the set of principal ideals generated by elements α ≡ 1 mod * m (i.e., α ≡ 1 mod m 0 and α (i) > 0 for all (.) (i) |m ∞ ) and finally by Cl m := I m /P m the ray class group modulo m. There are efficient algorithms for computing ray class groups [6, 17] provided we already know the unit and class group of k.An ideal group H (defined mod m) is a subgroup of I m containing P m . For any ideal group H letH := I m /H.

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Section: Comparisonmentioning

confidence: 99%

Computing class fields via the Artin map

Fieker

2000

Math. Comp.

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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: 90%

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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“…Here the regulator matrix is 1 by 1, and involves a single special unit, so we have the possibility of actually constructing this special unit from the first derivatives at 0 of certain Artin L-functions. This method of constructing S-units while simultaneously gaining numerical confirmation of the conjecture at hand appears in [Dummit et al 1997] and [Roblot 2000] for the abelian case. A difference in this paper is that the extension field K is no longer a class field which can be explicitly constructed from abelian L-functions by means of the conjecture.…”

Section: Introductionmentioning

confidence: 90%