I know that if $A$ is a simple abelian variety over a number field $k$ with all endomorphisms defined over $K$ then $\mbox{End}(A_K)\otimes \mathbb{Q}$ is a division algebra with a positive involution. Albert's classification tells us that these types of division algebras come in four types

Type I: Totally real number field

Type II/III: Central simple algebra $L$ over a totally real field such that simple components of $L\otimes \mathbb{R}$ are isomorphic to $M_2(\mathbb{R})$ or $\mathbb{H}$ (depending on Type II or III respectively)

Type IV: Central simple algebra over a CM-field.

My question is, for a fixed dimension, $g$, can we find simple abelian varieties of dimension $g$ such that it's endomorphism ring falls into each of these types?

I ask because in the paper Sato-Tate distributions and Galois endomorphism modules in genus 2 (by Fite, Kedlaya, Rotger and Sutherland), they say that for simple abelian varieties of dimension 2, then $\mbox{End}(A_K)\otimes \mathbb{R}$ can be one of

$$\mathbb{R}, \mathbb{R}\times\mathbb{R}, \mathbb{C}\times\mathbb{C}, M_2(\mathbb{R})$$

If it were true that for a fixed dimension all types appear, shouldn't it be possible to get an abelian variety with endomoprhism ring $\mathbb{H}$? If it is the case that not all types can appear for a fixed dimension, is it known when they can appear?


Albert's classification works/is good enough for (algebraically closed) fields of characteristic zero. For the complete list of all possibilities for a given $g$ (including the case of prime characteristic) see a survey of Frans Oort:

``Endomorphism algebras of abelian varieties". Algebraic geometry and commutative algebra, Vol. II, 469–502, Kinokuniya, Tokyo, 1988. (MR0977774 (90j:11049) ).

In particular, if $g=2$ then you cannot get type III.


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