Modèles minimaux des variétés abéliennes sur les corps locaux et globaux

Neron, André

doi:10.1007/bf02684271

Cited by 328 publications

(267 citation statements)

References 17 publications

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“…In F-theory, nonAbelian gauge groups appear when the elliptic fibration admits reducible singular fibers over a component of the discriminant locus [2,3]. Such singular fibers located over a codimension-one locus in the base of an elliptic fibration were classified by Kodaira and Néron and admit dual graphs that are extended ADE Dynkin diagrams [51,52]. Non-simply laced gauge groups can also be obtained by taking into account the monodromy action by an outer automorphism on the nodes of the dual graph of the singular fiber [3].…”

Section: 1mentioning

confidence: 99%

Tate form and weak coupling limits in F-theory

Esole

Savelli

2013

J. High Energ. Phys.

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AbsractWe consider the weak coupling limit of F-theory in the presence of non-Abelian gauge groups implemented using the traditional ansatz coming from Tate's algorithm. We classify the types of singularities that could appear in the weak coupling limit and explain their resolution. In particular, the weak coupling limit of SU(n) gauge groups leads to an orientifold theory which suffers from conifold singulaties that do not admit a crepant resolution compatible with the orientifold involution. We present a simple resolution to this problem by introducing a new weak coupling regime that admits singularities compatible with both a crepant resolution and an orientifold symmetry. We also comment on possible applications of the new limit to model building. We finally discuss other unexpected phenomena as for example the existence of several non-equivalent directions to flow from strong to weak coupling leading to different gauge groups.

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Section: 1mentioning

confidence: 99%

Tate form and weak coupling limits in F-theory

Esole

Savelli

2013

J. High Energ. Phys.

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“…Soit Q ∈ C, en reprenant la classification donnée par Néron [Nér64] et les notations de Tate [Tat75] on obtient :…”

Section: Majoration Du Discriminantunclassified

“…Pour cela on va regarder plus précisement le modèle régulier minimal à l'aide de sa construction par Néron [Nér64] à partir du modèle plan.…”

Section: Il Nous Reste Donc à éTudierunclassified

“…De manière récursive on construit un modèle comme joint des modèles (on corrigera une petite erreur dans [Nér64] Sect. 12)…”

Section: Caractéristiqueunclassified

“…Le cas isotrivial est donc un cas particulier du cas j entier. Or, lorsque j est entier et la caractéristique différente de 2, le nombre de composantes de la fibre spéciale associée à v est inférieur à 9 (toujours d'après [Nér64]), soit localement -si deg(D E/C ) 12s, étant donné qu'ici m P 9, l'égalité donne:…”

Section: Cas Où J Est Entierunclassified

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Pesenti¹,

Szpiro²

2000

Compositio Mathematica

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Abstract. In this article the degree of the discriminant of an elliptic pencil on a projective curve is upper-bounded by using the degree of its conductor and the genus of the base curve. This is done in the most general case, extending a method and a result of Szpiro (1981 and 1990a) and a result of Hindry and Silvermann. The difficult part, dealing with characteristic 2 and 3 and additive reductions, need the introduction of a new object -which we called 'conducteur efficace' -defined by using differentials and interestingly comparable to the usual conductor. This article ends with a few results in the arithmetical case -case corresponding to an inequality conjectured by the second author in 1978: (1) the proof of this inequality in the potentially good reduction cases; (2) the passage from the semi-stable reduction to the general case for a strong inequality.Mathematics Subject Classifications (1991): 11G05, 11G07.

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Class Groups and Selmer Groups

Schaefer

1996

Journal of Number Theory

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It is often the case that a Selmer group of an abelian variety and a group related to an ideal class group can both be naturally embedded into the same cohomology group. One hopes to compute one from the other by finding how close each is to their intersection. In this paper we compute the two groups and their intersection explicitly in the local case and put together the local information to get sharp upper bounds in the global case. The techniques in this paper can be used for arbitrary abelian varieties, isogenies and number fields assuming a frequently occurring condition. Several examples are worked out for the Jacobians of elliptic and hyperelliptic curves.

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Modèles minimaux des variétés abéliennes sur les corps locaux et globaux

Cited by 328 publications

References 17 publications

Tate form and weak coupling limits in F-theory

Tate form and weak coupling limits in F-theory

Untitled

Class Groups and Selmer Groups

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