Nous prouvons un théorème de finitude pour le premier groupe de cohomologie plate des schémas en groupes finis et plats sur les variétés intègres, normales et propres sur un corps fini. En conséquence, nous pouvons prouver l’invariance de la finitude de la partie -primaire du groupe de Tate–Shafarevich des schémas abéliens sur des bases de dimension supérieure par isogénie et changement de base. Chemin faisant, nous généralisons certains des résultats précédents sur le groupe de Tate–Shafarevich dans ce contexte.
We prove a finiteness theorem for the first flat cohomology group of finite flat group schemes over integral normal proper varieties over finite fields. As a consequence, we can prove the invariance of the finiteness of the Tate–Shafarevich group of Abelian schemes over higher dimensional bases under isogenies and alterations over/of such bases for the -part. Along the way, we generalize previous results on the Tate–Shafarevich group in this situation.
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Mots clés : Tate–Shafarevich groups of abelian varieties over higher dimensional bases over finite fields, $p$-torsion in characteristic $p > 0$; Abelian varieties of dimension $> 1$; Étale and other Grothendieck topologies and cohomologies; Arithmetic ground fields for abelian varieties
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@article{JTNB_2022__34_2_497_0,
author = {Timo Keller},
title = {On the $p$-torsion of the {Tate{\textendash}Shafarevich} group of abelian varieties over higher dimensional bases over finite fields},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {497--513},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {34},
number = {2},
year = {2022},
doi = {10.5802/jtnb.1211},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1211/}
}
TY - JOUR AU - Timo Keller TI - On the $p$-torsion of the Tate–Shafarevich group of abelian varieties over higher dimensional bases over finite fields JO - Journal de théorie des nombres de Bordeaux PY - 2022 SP - 497 EP - 513 VL - 34 IS - 2 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1211/ DO - 10.5802/jtnb.1211 LA - en ID - JTNB_2022__34_2_497_0 ER -
%0 Journal Article %A Timo Keller %T On the $p$-torsion of the Tate–Shafarevich group of abelian varieties over higher dimensional bases over finite fields %J Journal de théorie des nombres de Bordeaux %D 2022 %P 497-513 %V 34 %N 2 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1211/ %R 10.5802/jtnb.1211 %G en %F JTNB_2022__34_2_497_0
Timo Keller. On the $p$-torsion of the Tate–Shafarevich group of abelian varieties over higher dimensional bases over finite fields. Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 2, pp. 497-513. doi : 10.5802/jtnb.1211. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1211/
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