Class groups, totally positive units, and squares

Edgar, Hugh M.; Mollin, R. A.; Peterson, Brian L.

doi:10.1090/s0002-9939-1986-0848870-x

Cited by 10 publications

(8 citation statements)

References 13 publications

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“…We give an explicit counterexample (of smallest degree). Let K be the degree-27 normal closure over Q of the field K 0 of discriminant 3 16 • 37 4 defined by…”

Section: Structural Resultsmentioning

confidence: 99%

On unit signatures and narrow class groups of odd degree abelian number fields

Breen¹,

Varma²,

Voight³

et al. 2019

Preprint

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For an abelian number field of odd degree, we study the structure of its 2-Selmer group as a bilinear space and as a Galois module. We prove structural results and make predictions for the distribution of unit signature ranks and narrow class groups in families where the degree and Galois group are fixed. Contents 1. Introduction 2. Properties of the 2-Selmer group and its signature spaces 3. Galois module structure for invariants of odd Galois number fields 4. Structural results 5. Conjectures 6. Computations Appendix A. Cyclic cubic fields with signature rank 1 (with Noam Elkies) References

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“…We give an explicit counterexample (of smallest degree). Let K be the degree-27 normal closure over Q of the field K 0 of discriminant 3 16 • 37 4 defined by…”

Section: Structural Resultsmentioning

confidence: 99%

On unit signatures and narrow class groups of odd degree abelian number fields

Breen¹,

Varma²,

Voight³

et al. 2019

Preprint

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“…There is in fact a conjecture of Davis and Taussky that says that ρ ∞ = 0 in the case of L = Q(ζ q + ζ −1 q ), where p = (q − 1)/2 is a Sophie Germain prime. For more on the Davis-Taussky conjecture see [4], [6], [7], [11], and [18]. Conjecture 2.20 (Davis-Taussky conjecture).…”

Section: Theorem 218 ([7]mentioning

confidence: 99%

Bounds of the rank of the Mordell–Weil group of Jacobians of Hyperelliptic Curves

Daniels¹,

Lozano-Robledo²,

Wallace³

2020

Journal De Théorie Des Nombres De Bordeaux

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© Société Arithmétique de Bordeaux, 2020, tous droits réservés.L'accès aux articles de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.centre-mersenne.org/), implique l'accord avec les conditions générales d'utilisation (http://jtnb. centre-mersenne.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedramArticle mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.centre-mersenne.org/

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“…Furthermore, its possible group structures are known (see [Vig76]). The group (O × ) 2 \O × + is always finite, and equals the null group in many cases, such as for fields K having narrow class number equal to 1 (see [EMP86]).…”

Section: Note That If [Imentioning

confidence: 99%

Computing ideal classes representatives in quaternion algebras

Pacetti

Sirolli

2014

Math. Comp.

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Let $K$ be a totally real number field and let $B$ be a totally definite quaternion algebra over $K$. In this article, given a set of representatives for ideal classes for a maximal order in $B$, we show how to construct in an efficient way a set of representatives of ideal classes for any Bass order in $B$. The algorithm does not require any knowledge of class numbers, and improves the equivalence checking process by using a simple calculation with global units. As an application, we compute ideal classes representatives for an order of level 30 in an algebra over the real quadratic field $\Q[\sqrt{5}]$.Comment: Corrected version. Section 4 has been rewritten from scratc

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Class groups, totally positive units, and squares

Cited by 10 publications

References 13 publications

On unit signatures and narrow class groups of odd degree abelian number fields

On unit signatures and narrow class groups of odd degree abelian number fields

Bounds of the rank of the Mordell–Weil group of Jacobians of Hyperelliptic Curves

Computing ideal classes representatives in quaternion algebras

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