Dans la continuité de nos travaux précédents, nous étudions un analogue, pour le modèle de Néron d’une variété abélienne semi-stable sur un corps de nombres, du class-invariant homomorphism introduit par M. J. Taylor, qui nous permet de mesurer la structure galoisienne de certains torseurs.
This paper is devoted to the study of the volcanoes of ℓ-isogenies of elliptic curves over a finite field, focusing on their height as well as on the location of curves across its different levels. The core of the paper lies on the relationship between the ℓ-Sylow subgroup of an elliptic curve and the level of the volcano where it is placed. The particular case ℓ = 3 is studied in detail, giving an algorithm to determine the volcano of 3-isogenies of a given elliptic curve. Experimental results...
Let be a quadratic imaginary number field of discriminant . For let denote the order of conductor in and its modular invariant which is known to generate the ring class field modulo over . The coefficients of the minimal equation of being quite large Weber considered in [We] the functions defined below and thereby obtained simpler generators of the ring class fields. Later on the singular values of these functions played a crucial role in Heegner’s solution [He] of the class...
Consider a representation of a finite group as automorphisms of a power series ring over a perfect field of positive characteristic. Let be the associated formal mixed-characteristic deformation functor. Assume that the action of is weakly ramified, i.e., the second ramification group is trivial. Example: for a group action on an ordinary curve, the action of a ramification group on the completed local ring of any point is weakly ramified.We prove that the only such that are not pro-representable...
In this article we will describe a surprising observation that occurred in the construction of quadratic unramified extensions of a family of pure cubic number fields. Attempting to find an explanation will lead us on a magical mystery tour through the land of pure cubic number fields, Hilbert class fields, and elliptic curves.
B. Poonen a récemment exhibé des exemples de variétés projectives et lisses de dimension 3 sur un corps de nombres qui n’ont pas de point rationnel et pour lesquelles il n’y a pas d’obstruction de Brauer–Manin après revêtement fini étale. Je montre que les variétés qu’il construit possèdent des zéro-cycles de degré 1.