Abstract
We deal with g-dimensional abelian varieties X over finite fields. We prove that there is a universal constant (positive integer) N=N(g) 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.
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