Jean Gillibert

Invariants de classes : Le cas semi-stable

We define here an analogue, for a semi-stable group scheme whose generic fiber is an abelian variety, of M. J. Taylors class-invariant homomorphism (defined for abelian schemes), and we give a geometric description of it. Then we extend a result of Taylor, Srivastav, Agboola and Pappas concerning the kernel of this homomorphism in the case of an elliptic curve.

Chevalley–Weil theorem and subgroups of class groups

We prove, under some mild hypothesis, that an ´etale cover of curves defined over a number field has infinitely many specializations into an everywhere unramified extension of number fields. This constitutes an “absolute” version of the Chevalley–Weil theorem. Using this result, we are able to generalise the techniques of Mestre, Levin and the second author for constructing and counting number fields with large class group.

Invariants de classes : propriétés fonctorielles et applications à l'étude du noyau

The class-invariant homomorphism allows one to measure the Galois module structure of torsors--under a finite flat group scheme--which lie in the image of a coboundary map associated to an exact sequence. It has been introduced first by Martin Taylor (the exact sequence being given by an isogeny between abelian schemes). We begin by giving general properties of this homomorphism, then we pursue its study in the case when the exact sequence is given by the multiplication by

Category theory, logic and formal linguistics: Some connections, old and new

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Galois module structure and Jacobians of Fermat curves

on an extension of an abelian scheme by a torus.

Prolongement de biextensions et accouplements en cohomologie log plate

We seize the opportunity of the publication of selected papers from the Logic, categories, semantics workshop to survey some current trends in logic, namely intuitionistic and linear type theories, that interweave categorical, geometrical and computational considerations. We thereafter present how these rich logical frameworks can model the way language conveys meaning.

Invariants de classes : exemples de non-annulation en dimension supérieure

The class-invariant homomorphism allows one to measure the Galois module structure of extensions obtained by dividing points on abelian varieties. In this paper, we consider the case when the abelian variety is the Jacobian of a Fermat curve. We give examples of torsion points whose associated Galois structure is trivial, as well as points of infinite order whose associated Galois structure is non-trivial.

The class group pairing and p-descent on elliptic curves

Nous revisitons, dans le langage des log schemas, le probleme de prolongement de biextensions de schemas en groupes commutatifs lisses par le groupe multiplicatif etudie par Grothendieck dans [10]. Nous montrons que ce probleme admet en general une solution dans la categorie des faisceaux pour la topologie log plate, contrairement a ce que lon peut observer en topologie fppf pour laquelle Grothendieck a defini des obstructions monodromiques. En particulier, dans le cas dune variete abelienne et de sa duale, il est possible de prolonger la biextension de Weil sur la totalite des modeles de Neron; ceci permet de definir un accouplement sur les points qui combine laccouplement de classes defini par Mazur et Tate et laccouplement de monodromie.We study, using the language of log schemes, the problem of extending biextensions of smooth commutative group schemes by the multiplicative group. This was first considered by Grothendieck in [10]. We show that this problem admits a solution in the category of sheaves for Katos log flat topology, in contradistinction to what can be observed using the fppf topology, for which monodromic obstructions were defined by Grothendieck. In particular, in the case of an abelian variety and its dual, it is possible to extend the Weil biextension to the whole Neron model. This allows us to define a pairing on the points that combines the class group pairing defined by Mazur and Tate and Grothendiecks monodromy pairing.

Cohomologie log plate, actions modérées et structures galoisiennes

RésuméLe class-invariant homomorphism permet de mesurer la structure galoisienne des torseurs—sous un schéma en groupes fini et plat G—qui sont dans l’image du cobord associé à une isogénie, de noyau G, entre des (modèles de Néron de) variétés abéliennes. Quand les variétés sont des courbes elliptiques à réduction semi-stable et que l’ordre de G est premier à 6, on sait que cet homomorphisme s’annule sur les points de torsion. Dans cet article, en nous servant de restrictions de Weil de courbes elliptiques, nous construisons, pour tout nombre premier pxa0>xa02, une variété abélienne A de dimension p munie d’une isogénie (de noyauxa0μp) dont le cobord est surjectif. Si A est de rang nul, et si la p-partie du groupe de Picard de la base est non triviale, nous obtenons ainsi un exemple où le class-invariant homomorphism ne s’annule pas sur les points de torsion.AbstractThe so-called class-invariant homomorphism ψ measures the Galois module structure of torsors—under a finite flat group scheme G—which lie in the image of a coboundary map associated to an isogeny between (Néron models of) abelian varieties with kernel G. When the varieties are elliptic curves with semi-stable reduction and the order of G is coprime to 6, it is known that the homomorphism ψ vanishes on torsion points. In this paper, using Weil restrictions of elliptic curves, we give the construction, for any prime number pxa0>xa02, of an abelian variety A of dimension p endowed with an isogeny (with kernelxa0μp) whose coboundary map is surjective. In the case when A has rank zero and the p-part of the Picard group of the base is non-trivial, we obtain examples where ψ does not vanish on torsion points.

Pulling back torsion line bundles to ideal classes

We give explicit formulae for the logarithmic class group pairing on an elliptic curve defined over a number field. Then we relate it to the descent relative to a suitable cyclic isogeny. This allows us to connect the resulting Selmer group with the logarithmic class group of the base. These constructions are explicit and suitable for computer experimentation. From a conceptual point of view, the questions that arise here are analogues of visibility questions in the sense of Cremona and Mazur.