Non-supersingular hyperelliptic jacobians

Zarhin, Yuri G.

Non-supersingular hyperelliptic jacobians
[Jacobiennes hyperelliptiques non supersingulières]

Zarhin, Yuri G.

Bulletin de la Société Mathématique de France, Tome 132 (2004), p. 617-634 / Harvested from Numdam

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Résumé
Abstract

Soient $K$ un corps de caractéristique impaire $p$ et $f (x)$ un polynôme irréductible séparable dans $K [x]$ de degré $n \geq 5$ , avec grand groupe de Galois (le groupe symétrique ou le groupe alterné). Soit $C$ la courbe hyperelliptique $y^{2} = f (x)$ et $J (C)$ sa jacobienne. Nous montrons que $J (C)$ n’a pas d’endomorphisme non trivial sur une clôture algébrique de $K$ si $n \geq 7$ ou $p \neq 3$ .

Let $K$ be a field of odd characteristic $p$ , let $f (x)$ be an irreducible separable polynomial of degree $n \geq 5$ with big Galois group (the symmetric group or the alternating group). Let $C$ be the hyperelliptic curve $y^{2} = f (x)$ and $J (C)$ its jacobian. We prove that $J (C)$ does not have nontrivial endomorphisms over an algebraic closure of $K$ if either $n \geq 7$ or $p \neq 3$ .

Publié le : 2004-01-01

MR 2131907 | Zbl 1079.14038

DOI : https://doi.org/10.24033/bsmf.2477
Classification: 14H40, 14K05
Mots clés: jacobiennes hyperelliptiques, endomorphismes des variétés abéliennes, variétés abéliennes supersingulières

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     author = {Zarhin, Yuri},
     title = {Non-supersingular hyperelliptic jacobians},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     volume = {132},
     year = {2004},
     pages = {617-634},
     doi = {10.24033/bsmf.2477},
     mrnumber = {2131907},
     zbl = {1079.14038},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BSMF_2004__132_4_617_0}
}

Zarhin, Yuri G. Non-supersingular hyperelliptic jacobians. Bulletin de la Société Mathématique de France, Tome 132 (2004) pp. 617-634. doi : 10.24033/bsmf.2477. http://gdmltest.u-ga.fr/item/BSMF_2004__132_4_617_0/

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