The logarithmic class group package in PARI/GP

Belabas, Karim; Jaulent, Jean-François

doi:10.5802/pmb.o-1

Cited by 26 publications

(69 citation statements)

References 20 publications

(19 reference statements)

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“…Our purpose has nothing to do with computational or theoretical approaches in the area of the "main theorem" on abelian fields (analytic formulas, cyclotomic units, L p -functions, etc.) as, for instance, the very many contributions (cited in our papers [5,6]), also giving computations and suggesting that equidistribution results may have striking consequences for the conjecture; our viewpoint is essentially logical and based on the governing group T k , because we have conjectured that T k = 1 for all p ≫ 0, due to 1 For more information on the main pioneering works about the practice of this theory, see "history of abelian p-ramification" in [9, Appendix] (e.g., Gras: "Crelle's Journal" (1982/83), Jaulent: "Ann. Inst.…”

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confidence: 84%

“…We call Greenberg's conjecture for k and p, the nullity of the Iwasawa invariants λ, µ (see the origin of the conjecture in [10,Theorems 1 and 2]). The main effective test for this conjecture is the criterion of Jaulent [14,Théorèmes A,B] proving that the conjecture is equivalent to the capitulation in k ∞ of the logarithmic class group C k of k (defined in [12] with PARI/GP pakage in [1]), an invariant also related to S-ramification theory. 1 For specific cases of decomposition of p, as in [10], see [19].…”

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confidence: 99%

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Algorithmic complexity of Greenberg’s conjecture

Gras¹

2021

Arch. Math.

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Let k be a totally real number field and p a prime. We show that the "complexity" of Greenberg's conjecture (λ = µ = 0) is of padic nature governed (under Leopoldt's conjecture) by the finite torsion group T k of the Galois group of the maximal abelian p-ramified pro-pextension of k, by means of images in T k of ideal norms from the layers k n of the cyclotomic tower (Theorem 5.2). These images are obtained via the formal algorithm computing, by "unscrewing", the p-class group of k n . Conjecture 5.4 of equidistribution of these images would show that the number of steps b n of the algorithms is bounded as n → ∞, so that Greenberg's conjecture, hopeless within the sole framework of Iwasawa's theory, would hold true "with probability 1". No assumption is made on [k : Q], nor on the decomposition of p in k/Q.

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confidence: 84%

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confidence: 99%

Algorithmic complexity of Greenberg’s conjecture

Gras¹

2021

Arch. Math.

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“…In the similar context of p-ramification, a new PARI/GP package allows the computation of the logarithmic class group C K of a number field by Belabas-Jaulent [86] that we can illustrate as follows where the invariants [Y, X, Z] are linked by the exact sequence:…”

Section: A65 Fermat Curvesmentioning

confidence: 99%

Practice of the Incomplete $p$-Ramification Over a Number Field -- History of Abelian $p$-Ramification

Gras¹

2019

Communications in Advanced Mathematical Sciences

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The theory of p-ramification, regarding the Galois group of the maximal pro-p-extension of a number field K, unramified outside p and ∞, is well known including numerical experiments with PARI/GP programs. The case of "incomplete p-ramification" (i.e., when the set S of ramified places is a strict subset of the set P of the p-places) is, on the contrary, mostly unknown in a theoretical point of view. We give, in a first part, a way to compute, for any S ⊆ P, the structure of the Galois group of the maximal S-ramified abelian pro-p-extension H K,S of any field K given by means of an irreducible polynomial. We publish PARI/GP programs usable without any special prerequisites. Then, in an Appendix, we recall the "story" of abelian S-ramification restricting ourselves to elementary aspects in order to precise much basic contributions and references, often disregarded, which may be used by specialists of other domains of number theory. Indeed, the torsion T K,S of Gal(H K,S /K) (even if S = P) is a fundamental obstruction in many problems. All relationships involving S-ramification, as Iwasawa's theory, Galois cohomology, p-adic L-functions, elliptic curves, algebraic geometry, would merit special developments, which is not the purpose of this text. Practice of the Incomplete p-Ramification Over a Number Field -History of Abelian p-Ramification -252/280The analogous theory for a local base field has also a long history that we shall not consider in this article.

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“…As an arithmetic invariant of a number field the logarithmic class group has importance in its own. The logarithmic class group is effective by computational methods [3,2,1] and it is related to the wild kernels in K-theory [14]. In spite of its tight relation to the ℓ-class group Cℓ K of K, it behaves differently in several situations (see §5).…”

Section: Introductionmentioning

confidence: 99%

On the mu and lambda invariants of the logarithmic class group

Villanueva-Gutiérrez

2020

Journal of Number Theory

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Let ℓ be a rational prime number. Assuming the Gross-Kuz'min conjecture along a Z ℓ-extension K∞ of a number field K, we show that there exist integers µ, λ and ν such that the exponentẽn of the order ℓẽ n of the logarithmic class group Cℓn for the n-th layer Kn of K∞ is given byẽn = µℓ n + λn + ν, for n big enough. We show some relations between the classical invariants µ and λ, and their logarithmic counterparts µ and λ for some class of Z ℓ-extensions. Additionally, we provide numerical examples for the cyclotomic and the non-cyclotomic case. * K ≃ Gal(K lc /K) is a noetherian Z ℓ-module, and assuming the Gross-Kuz'min Conjecture it has rank 1. This yields the following result.

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The logarithmic class group package in PARI/GP

Cited by 26 publications

References 20 publications

Algorithmic complexity of Greenberg’s conjecture

Algorithmic complexity of Greenberg’s conjecture

Practice of the Incomplete $p$-Ramification Over a Number Field -- History of Abelian $p$-Ramification

On the mu and lambda invariants of the logarithmic class group

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