Non-supersingular hyperelliptic jacobians
Bulletin de la Société Mathématique de France, Volume 132 (2004) no. 4, p. 617-634

Let K be a field of odd characteristic p, let f(x) be an irreducible separable polynomial of degree n5 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 n7 or p3.

Soient K un corps de caractéristique impaire p et f(x) un polynôme irréductible séparable dans K[x] de degré n5, 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 n7 ou p3.

DOI : https://doi.org/10.24033/bsmf.2477
Classification:  14H40,  14K05
Keywords: hyperelliptic jacobians, endomorphisms of abelian varieties, supersingular abelian varieties
@article{BSMF_2004__132_4_617_0,
     author = {Zarhin, Yuri},
     title = {Non-supersingular hyperelliptic jacobians},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {132},
     number = {4},
     year = {2004},
     pages = {617-634},
     doi = {10.24033/bsmf.2477},
     zbl = {1079.14038},
     mrnumber = {2131907},
     language = {en},
     url = {http://www.numdam.org/item/BSMF_2004__132_4_617_0}
}
Zarhin, Yuri G. Non-supersingular hyperelliptic jacobians. Bulletin de la Société Mathématique de France, Volume 132 (2004) no. 4, pp. 617-634. doi : 10.24033/bsmf.2477. http://www.numdam.org/item/BSMF_2004__132_4_617_0/

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