There are genus one curves of every index over every infinite, finitely generated field

Clark, Pete L.; Lacy, Allan

doi:10.1515/crelle-2016-0037

Cited by 6 publications

(4 citation statements)

References 27 publications

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“…Corollary 6 has many applications. For instance, if k is a number field, there are infinitely many smooth quintic genus one curves in P 3 defined over k that do not have k-rational points [3,4,11]. Thus, using Corollary 6 and Pfaffian representations of quintic elliptic curves [5], one can construct infinitely many explicit examples of K-stable smooth Fano threefolds in the family No 2.17.…”

Section: Corollary 5 If the Intersection Ementioning

confidence: 99%

On K-stability of $$\mathbb {P}^3$$ blown up along a quintic elliptic curve

Cheltsov,

Pokora

2024

Ann Univ Ferrara

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Section: Corollary 5 If the Intersection Ementioning

confidence: 99%

On K-stability of $$\mathbb {P}^3$$ blown up along a quintic elliptic curve

Cheltsov,

Pokora

2024

Ann Univ Ferrara

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“…Recall that is the rational prime below p. By the classification theorem of compact -adic Lie groups (cf. [4,Theorem 21]):…”

Section: Inertia Groups Over = Pmentioning

confidence: 99%

On class numbers of division fields of abelian varieties

Garnek¹

2019

J. Théor. Nombres Bordeaux

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L'accès aux articles de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/), implique l'accord avec les conditions générales d'utilisation (http://jtnb.cedram. 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. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Journal de Théorie des Nombres de Bordeaux 31 (2019), 227-242 On class numbers of division fields of abelian varieties par Jędrzej GARNEK Résumé. Soit A une variété abélienne définie sur un corps de nombres K. On fixe un nombre premier p et pour tout nombre naturel n, on note K n le corps engendré sur K par les coordonnées des points de p n-torsion de A. Nous donnons une minoration de l'ordre de la p-partie du groupe de classes de K n pour n 0, en construisant une extension non ramifiée suffisamment grande de K n. Cette minoration dépend du rang du groupe de Mordell-Weil de A et de la réduction des points de p-torsion en nombres premiers au-dessus de p.

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“…For results on period and index over more general fields, see [CL15]. Applications of the period and index pop up typically in relation to the size of Shafarevich-Tate groups; see for example [PS99] and [LLR04].…”

Section: Period and Indexmentioning

confidence: 99%

Period and index for higher genus curves

Sharif

2018

Journal of Number Theory

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Period and indexLet K be a field; usually, we will consider it to be a number field. By a curve C over K, we shall mean a smooth projective geometrically integral curve. The period and index of C over K are two integer invariants which measure the failure of C to have rational points. Specifically, the index is the gcd of degrees of all effective divisors D ∈ Div C-that is, effective divisors which are rational over K. Equivalently, the index is the gcd of degreesσP is a rational effective divisor, where σ ranges over the embeddings of L into an algebraic closure of K; and conversely any minimal rational effective divisor is of this form. Also observe that if C already has K-points, then its index is 1.The period is less stringent: here we look at the smallest positive degree of rational divisor classes; these are given by divisors which are linearly equivalent to their Galois conjugates. To see that the two invariants need not be the same, consider any conic over R without rational points, say the curve with affine piece x 2 + y 2 = −1. Certainly the index is 2, but as our curve is genus 0, there is a single divisor class of degree 1; namely, the class of a point. Therefore this class must be rational over R, and so the period is 1.As a rational divisor automatically belongs to a rational divisor class, we have P | I. Furthermore, the canonical class gives a rational divisor of degree 2(g − 1), so I | 2(g − 1). Lichtenbaum in [Lic69] found further conditions on the possible values of the period and index: Theorem 1.1 (Theorem 8,[Lic69]). Let C/K be a curve over a field with genus g, period P , and index I. Then2 , and(ii) if either 2(g − 1)/I or P is even, then I | P 2 .We say a triple of integers (g, P, I) is admissible if they satisfy the divisibility conditions of Lichtenbaum's theorem; that is, if they are possible values for the genus, period, and index of a curve. The goal of this paper is to determine whether every admissible triple indeed occurs as the invariants of some curve over a number field. Our main result is the following: 1

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There are genus one curves of every index over every infinite, finitely generated field

Cited by 6 publications

References 27 publications

On K-stability of $$\mathbb {P}^3$$ blown up along a quintic elliptic curve

On K-stability of $$\mathbb {P}^3$$ blown up along a quintic elliptic curve

On class numbers of division fields of abelian varieties

Period and index for higher genus curves

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