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Valeur en $2$ de fonctions $L$ de formes modulaires de poids $2$ : théorème de Beilinson explicite

François Brunault (2007)

Bulletin de la Société Mathématique de France

Nous montrons une version explicite du théorème de Beilinson pour la courbe modulaire $X_{1} (N)$ . Ce résultat est la première étape d’un travail reliant, d’une part, la valeur en $2$ de la fonction $L$ d’une forme primitive de poids $2$ , et d’autre part, la fonction dilogarithme associée à la courbe modulaire correspondante, dans l’esprit de la conjecture de Zagier pour les courbes elliptiques. Comme corollaire de notre théorème, dans le cas où $N$ est premier, nous répondons à une question de Schappacher et Scholl...

Variation of periods modulo $p$ in arithmetic dynamics.

Silverman, Joseph H. (2008)

The New York Journal of Mathematics [electronic only]

Variation of the reduction type of elliptic curves under small base change with wild ramification

Masanari Kida (2003)

Open Mathematics

We study the variation of the reduction type of elliptic curves under base change. A complete description of the variation is given when the base field is the p-adic field and the base change is of small degree.

Variation of the root number in families of elliptic curves

David E. Rohrlich (1993)

Compositio Mathematica

Variations on a theme of Runge: effective determination of integral points on certain varieties

Aaron Levin (2008)

Journal de Théorie des Nombres de Bordeaux

We consider some variations on the classical method of Runge for effectively determining integral points on certain curves. We first prove a version of Runge’s theorem valid for higher-dimensional varieties, generalizing a uniform version of Runge’s theorem due to Bombieri. We then take up the study of how Runge’s method may be expanded by taking advantage of certain coverings. We prove both a result for arbitrary curves and a more explicit result for superelliptic curves. As an application of our...

Variétés abéliennes et invariants arithmétiques

Jean Gillibert (2006)

Annales de l’institut Fourier

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.

Variétés de Drinfeld compactes

Henri Carayol (1991/1992)

Séminaire Bourbaki

Visibility of Mordell-Weil groups.

Stein, William A. (2007)

Documenta Mathematica

Visibility of the Shafarevich-Tate group at higher level.

Jetchev, Dimitar P., Stein, William A. (2007)

Documenta Mathematica

Visualizing elements in the Shafarevich-Tate group.

Cremona, John E., Mazur, Barry (2000)

Experimental Mathematics

Volcanoes of l-isogenies of elliptic curves over finite fields: The case l=3.

Josep M. Miret Biosca, Daniel Sadornil Renedo, Juan Tena Ayuso, Rosana Tomàs, Magda Valls Marsal (2007)

Publicacions Matemàtiques

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...

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