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*?



Albert classification of rational endomorphism rings of simple Abelian varieties over finite fields
See <http://www.math.nyu.edu/~tschinke/books/finite-fields/final/05_oort.pdf>, Section 15:


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


