Sur le rang des jacobiennes sur un corps de fonctions

Hindry, Marc

doi:10.24033/bsmf.2487

Cited by 10 publications

(12 citation statements)

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“…Nous prouvons ici une formule (Proposition 2.6) analogue à celle de Shioda-Tate pour les surfaces elliptiques ayant une section généralisant [20,26,9]. Nous utilisons la théorie d'intersection et prenons comme référence Fulton [6] ou Hartshorne [8, Appendix A].…”

Section: Outils Géométriquesunclassified

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Fibrations et conjecture de Tate

Hindry

Wazir

2005

Journal of Number Theory

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Section: Outils Géométriquesunclassified

“…Hindry et Pacheco[9, Proposition 3.4]). Le groupe ker(f * ) est fini et est même trivial si f admet une section générique.…”

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Fibrations et conjecture de Tate

Hindry

Wazir

2005

Journal of Number Theory

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“…The K/k-trace of J is a quotient of the Picard variety (Cf. [HiPa05,Sh92,Sh99]) and hence is trivial.…”

Section: Shioda's Family Of Hyperelliptic Curvesmentioning

confidence: 95%

“…Proof. (sketch and references) See [Sh92] or [HiPa05] or, for a more general setting, see [HiPaWa05]. The main point is to define the map Pic(S) −→ Jac Xη (k(B)) and determine its kernel.…”

Section: Examplesmentioning

confidence: 99%

Lower bounds for the rank of families of abelian varieties under base change

Salgado

Hindry²

2019

Acta Arith.

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We consider the following question : given a family of abelian varieties A over a curve B defined over a number field k, how does the rank of the Mordell-Weil group of the fibres At(k) vary? A specialisation theorem of Silverman guarantees that, for almost all t in C(k), the rank of the fibre is at least the generic rank, that is the rank of A(k(B)). When the base curve B is rational, we show, at least in many cases and under some geometric conditions, that there are infinitely many fibres for which the rank is larger than the generic rank. This paper is a sequel to a paper of the second author [Sa12] where the case of elliptic surfaces is treated.

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“…Various versions of Proposition 4.2.1 appear in the literature, notably in [61], [15], [57], and [20], and as Shioda notes [57, p. 359], it was surely known to the ancients.…”

Section: Shioda-tate Theoremmentioning

confidence: 99%

Curves and Jacobians over Function Fields

Ulmer

2014

Advanced Courses in Mathematics - CRM Barcelona

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These notes originated in a 12-hour course of lectures given at the Centre de Recerca Mathemàtica near Barcelona in February, 2010. The aim of the course was to explain results on curves and their Jacobians over function fields, with emphasis on the group of rational points of the Jacobian, and to explain various constructions of Jacobians with large Mordell-Weil rank.More so than the lectures, these notes emphasize foundational results on the arithmetic of curves and Jacobians over function fields, most importantly the breakthrough works of Tate, Artin, and Milne on the conjectures of Tate, Artin-Tate, and Birch and Swinnerton-Dyer. We also discuss more recent results such as those of Kato and Trihan. Constructions leading to high ranks are only briefly reviewed, because they are discussed in detail in other recent and forthcoming publications.These notes may be viewed as a continuation of my Park City notes [69]. In those notes, the focus was on elliptic curves and finite constant fields, whereas here we discuss curves of high genera and results over more general base fields.It is a pleasure to thank the organizers of the CRM Research Program, especially Francesc Bars, for their efficient organization of an exciting meeting, the audience for their interest and questions, the National Science Foundation for funding to support the travel of junior participants (grant DMS 0968709), and the editors for their patience in the face of many delays. It is also a pleasure to thank Marc Hindry, Nick Katz, and Dino Lorenzini for their help. Finally, thanks to Timo Keller and René Pannekoek for corrections.I welcome all comments, and I plan to maintain a list of corrections and supplements on my web page. Please check there for updates if you find the material in these notes useful.Atlanta, Unless explicitly stated otherwise, all schemes are assumed to be Noetherian and separated and all morphisms of schemes are assumed to be separated and of finite type.A curve over a field F is a scheme over F that is reduced and purely of dimension 1, and a surface is similarly a scheme over F which is reduced and purely of dimension 2. Usually our curves and surfaces will be subject to further hypotheses, like irreducibility, projectivity, or smoothness.We recall that a scheme Z is regular if each of its local rings is regular. This means that for each point z ∈ Z, with local ring O Z,z , maximal ideal m z ⊂ O Z,z , and residue field κ(z) = O Z,z /m z , we haveEquivalently, m z should be generated by dim z Z elements. A morphism f : Z → S is smooth (of relative dimension n) at z ∈ Z if there exist affine open neighborhoods U of z and V of f (z) such that f (U ) ⊂ V and a diagram

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Sur le rang des jacobiennes sur un corps de fonctions

Cited by 10 publications

References 10 publications

Fibrations et conjecture de Tate

Fibrations et conjecture de Tate

Lower bounds for the rank of families of abelian varieties under base change

Curves and Jacobians over Function Fields

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