Nous donnons une famille infinie de courbes de genre 2 dont la Jacobienne possède des éléments non triviaux du groupe de Tate-Shafarevich pour une descente via l’isogénie de Richelot. Nous le prouvons en effectuant une descente via l’isogénie de Richelot et une 2-descente complète sur la Jacobienne isogène. Nous donnons également un modèle explicite d’une famille associée de surfaces qui violent le principe de Hasse.
We give an infinite family of curves of genus 2 whose Jacobians have non-trivial members of the Tate-Shafarevich group for descent via Richelot isogeny. We prove this by performing a descent via Richelot isogeny and a complete 2-descent on the isogenous Jacobian. We also give an explicit model of an associated family of surfaces which violate the Hasse principle.
@article{JTNB_2009__21_1_1_0,
author = {Arnth-Jensen, Anna and Flynn, E. Victor},
title = {Non-trivial $\Sha$ in the {Jacobian} of an infinite family of curves of genus 2},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {1--13},
publisher = {Universit\'e Bordeaux 1},
volume = {21},
number = {1},
year = {2009},
doi = {10.5802/jtnb.654},
zbl = {1179.14030},
mrnumber = {2537700},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jtnb.654/}
}
TY - JOUR AU - Arnth-Jensen, Anna AU - Flynn, E. Victor TI - Non-trivial $\Sha$ in the Jacobian of an infinite family of curves of genus 2 JO - Journal de théorie des nombres de Bordeaux PY - 2009 SP - 1 EP - 13 VL - 21 IS - 1 PB - Université Bordeaux 1 UR - http://www.numdam.org/articles/10.5802/jtnb.654/ DO - 10.5802/jtnb.654 LA - en ID - JTNB_2009__21_1_1_0 ER -
%0 Journal Article %A Arnth-Jensen, Anna %A Flynn, E. Victor %T Non-trivial $\Sha$ in the Jacobian of an infinite family of curves of genus 2 %J Journal de théorie des nombres de Bordeaux %D 2009 %P 1-13 %V 21 %N 1 %I Université Bordeaux 1 %U http://www.numdam.org/articles/10.5802/jtnb.654/ %R 10.5802/jtnb.654 %G en %F JTNB_2009__21_1_1_0
Arnth-Jensen, Anna; Flynn, E. Victor. Non-trivial $\Sha$ in the Jacobian of an infinite family of curves of genus 2. Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 1, pp. 1-13. doi : 10.5802/jtnb.654. http://www.numdam.org/articles/10.5802/jtnb.654/
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