Géométrie algébrique

Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces

[Vérification exacte de la conjecture BSD forte pour certaines variétés abéliennes absolument simples]

Keller, Timo¹   ; Stoll, Michael¹ 

¹ Lehrstuhl Mathematik II (Computeralgebra), Universität Bayreuth, Universitätsstraße 30, 95440 Bayreuth, Germany

Comptes Rendus. Mathématique, Tome 360 (2022) no. G5, pp. 483-489.

Abstract (VO)

Let $X$ be one of the $28$ Atkin–Lehner quotients of a curve $X_0\left(N\right)$ such that $X$ has genus $2$ and its Jacobian variety $J$ is absolutely simple. We show that the Shafarevich–Tate group $Ш(J/\mathbb{Q})$ is trivial. This verifies the strong BSD conjecture for $J$.

Résumé

Soit $X$ un des $28$ quotients d’Atkin–Lehner d’une courbe $X_0\left(N\right)$ tel que $X$ est de genre $2$ et sa jacobienne $J$ est absolument simple. On démontre que le groupe de Shafarevich–Tate $Ш(J/\mathbb{Q})$ est trivial. Ceci vérifie la conjecture BSD forte pour $J$.

Détail

Reçu le : 2021-06-30
Révisé le : 2021-11-05
Accepté le : 2021-12-09
Publié le : 2022-05-23

DOI : 10.5802/crmath.313

Classification : 11G40, 11-04, 11G10, 11G30, 14G35

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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