Question

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?

Answer 1

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 $2$.