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We give a necessary and sufficient condition in terms of a matrix for whichall Tate classes are Lefschetz for simple abelian varieties over an algebraicclosure of a finite field. As an application, we prove under an assumption thatall Tate classes are Lefschetz for simple factors of Fermat Jacobian of primedegree.

Lefschetz classes of simple factors of Fermat Jacobian of prime degree over finite fields