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.
Révisé le :
Accepté le :
Publié le :
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
@article{JTNB_2022__34_2_497_0,
author = {Keller, Timo},
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 = {http://archive.numdam.org/articles/10.5802/jtnb.1211/}
}
TY - JOUR AU - Keller, Timo 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 - http://archive.numdam.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 Keller, Timo %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 http://archive.numdam.org/articles/10.5802/jtnb.1211/ %R 10.5802/jtnb.1211 %G en %F JTNB_2022__34_2_497_0
Keller, Timo. 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. http://archive.numdam.org/articles/10.5802/jtnb.1211/
[1] Séminaire de Géométrie Algébrique du Bois Marie – 1963–64 – Théorie des topos et cohomologie étale des schémas – (SGA 4) – Vol. 2, Lecture Notes in Mathematics, 270, Springer, 1972, iv+418 pages
[2] Séminaire de Géométrie Algébrique du Bois Marie – 1966–67 – Théorie des intersections et théorème de Riemann-Roch – (SGA 6), Lecture Notes in Mathematics, 225, Springer, 1971, xii+700 pages
[3] Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 21, Springer, 1990, x+325 pages | DOI
[4] Algebraic geometry I. Schemes. With examples and exercises, Advanced Lectures in Mathematics, Vieweg+Teubner, 2010, vii+615 pages | DOI | Zbl
[5] Séminaire de Géométrie Algébrique du Bois Marie – 1960–61 – Revêtements étales et groupe fondamental – (SGA 1), Lecture Notes in Mathematics, 224, Springer, 1971, xxii+447 pages
[6] Eléments de géométrie algébrique : III. Étude cohomologique des faisceaux cohérents, Première partie., Publ. Math., Inst. Hautes Étud. Sci., Volume 11 (1961), pp. 5-167
[7] Algebraic geometry. Corr. 3rd printing., Graduate Texts in Mathematics, 52, Springer, 1983, xvi+496 pages
[8] Finite locally free group schemes in characteristic and Dieudonné modules., Invent. Math., Volume 114 (1993) no. 1, pp. 89-137 | DOI | Zbl
[9] Smoothness, semi-stability and alterations., Publ. Math., Inst. Hautes Étud. Sci., Volume 83 (1996), pp. 51-93 | Numdam | Zbl
[10] On the Tate–Shafarevich group of Abelian schemes over higher dimensional bases over finite fields, Manuscr. Math., Volume 150 (2016), pp. 211-245 | DOI | MR | Zbl
[11] On an analogue of the conjecture of Birch and Swinnerton–Dyer for Abelian schemes over higherdimensional bases over finite fields, Doc. Math., Volume 24 (2019), pp. 915-993 | MR | Zbl
[12] Algebraic geometry and arithmetic curves. Transl. by Reinie Erné., Oxford Graduate Texts in Mathematics, 6, Oxford University Press, 2006, xv+577 pages
[13] Étale cohomology., Princeton Mathematical Series, 33, Princeton University Press, 1980, xiii+323 pages | MR
[14] Abelian varieties, Arithmetic geometry (Pap. Conf., Storrs/Conn., 1984), Springer, 1986, pp. 103-150 | DOI | Zbl
[15] Abelian varieties, Oxford University Press, 1970, viii+242 pages
[16] Critères de platitude et de projectivité. Techniques de « platification » d’un module, Invent. Math., Volume 13 (1971), pp. 1-89 | DOI | Zbl
[17] Group schemes, formal groups, and p-divisible groups, Arithmetic geometry (Pap. Conf., Storrs/Conn., 1984), Springer, 1986, pp. 29-78 | DOI | MR | Zbl
[18] Galois groups and fundamental groups., Cambridge Studies in Advanced Mathematics, 117, Cambridge University Press, 2009, ix+270 pages | DOI
[19] Finitness of syntomic/fppf cohomology with coefficients in a finite flat group scheme MathOverflow, https://mathoverflow.net/q/257441 (version: 2016-12-19)
Cité par Sources :
