Tate classes on self-products of Abelian varieties over finite fields

Zarhin, Yuri G.

Annales de l'Institut Fourier, Tome 72 (2022) no. 6, pp. 2339-2383.

Nous étudions des variétés abéliennes $X$ de dimension $g$ sur des corps finis. Nous prouvons l’existence d’une constante universelle (entière positive) $N=N\left(g\right)$, qui ne dépend que de $g$ et a la propriété suivante : si une certaine puissance de $X$ admet une classe de Tate exotique, la puissance $X^{2N}$ de $X$ admet une classe de Tate exotique aussi. Cela donne une réponse positive à une question de Kiran Kedlaya.

We deal with $g$-dimensional abelian varieties $X$ over finite fields. We prove that there is a universal constant (positive integer) $N=N\left(g\right)$ that depends only on $g$ that enjoys the following property. If a certain self-product of $X$ carries an exotic Tate class then the self-product $X^{2N}$ of $X$ also carries an exotic Tate class. This gives a positive answer to a question of Kiran Kedlaya.

Classification : 11G10, 11G25, 14G15
Keywords: Abelian varieties, Tate classes, Finite fields
Mot clés : Variétés abéliennes, classes de Tate, corps finis

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