Non-supersingular hyperelliptic jacobians

Zarhin, Yuri G.

Bulletin de la Société Mathématique de France, Tome 132 (2004) no. 4, pp. 617-634.

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

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

Classification : 14H40, 14K05
Keywords: hyperelliptic jacobians, endomorphisms of abelian varieties, supersingular abelian varieties
Mot clés : jacobiennes hyperelliptiques, endomorphismes des variétés abéliennes, variétés abéliennes supersingulières

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