# Albert classification of rational endomorphism rings of simple Abelian varieties over finite fields

Recall the Albert classification of rational endomorphism rings with involution of simple Abelian varieties over arbitrary fields:

• Type I: totally real, trivial involution

• Type II and III: quaternion algebras over totally real number fields

• Type IV: center is a CM field for which the restriction of the involution is complex conjugation

Now my question is: Which of these division algebras can actually occur if the base field is a finite field?

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Only Type III and IV can occur, and III only for dimension $1$ or $2$.