Definitions of CM abelian varieties

Question

I was reading through a presentation by Oort (https://www2.math.upenn.edu/~chai/UPenn2013-beamer.pdf) and noticed something which disturbed me: he defines (slide 38) a simple CM abelian variety to be one whose endomorphism algebra contains a field of degree $2g$. He then says that the endomorphism algebra of a simple CM abelian variety need not be a field.

The definition I'm familiar with is that the reduced degree of the endomorphism algebra is $2g$. From this, one can show that a simple abelian variety is of CM type iff its endomorphism algebra is a degree $2g$ CM field. [EDIT: this is only true over characteristic zero fields! This was the main source of my confusion.]

The difference between the definitions is that Oort doesn't require the field to be the center of the endomorphism algebra. However, I can't think of an example where there is a degree $2g$ field properly contained in an endomorphism algebra of a simple abelian variety.

Hence my two questions:

1. What would be an example of a simple abelian variety which is CM in the sense of Oort, but whose endomorphism algebra is not a field?
2. Is it important to make a distinction between these two definitions, and which is the "standard" one? For instance, classification results exist for isogeny classes of CM abelian varieties, and the above seems to suggest these would differ depending on the notion of CM.

Update: in 4.2 of this book by Oort, he says that for abelian varieties over characteristic zero fields, his definition implies that the endomorphism algebra of a simple CM abelian variety is a degree $2g$ CM field, but not in positive characteristic. He says that there are many counterexamples in positive characteristic, but as far as I can see doesn't give an explicit one.

Answer 1

As you said in the update in characteristic zero two definitions are the same but In positive characteristics already in the case of elliptic curves, there are elliptic curves whose endomorphism rings are quaternion algebras which are skew field of $\mathbb{Q}$ dimension 4 and contains quadratic imaginary numberfields. In fact, an elliptic curve is supersingular if and only if its endomorphism ring is a quaternion algebra. for a good analysis of endomorphism of elliptic curves (the simplest case of abelian varieties but most of the interesting thing already happen in this case), you can read the arithmetic of elliptic curves by Silverman chapter 3 section 9 and also section 3 of chapter 5.

Also, Ort's definition is standard. you can learn more about CM multiplication from Milne notes which proofs the main theorems of CM multiplication using this general definition.

Answer 2

I think the most popular current definition is, as you say, that the reduced degree of the endomorphism algebra is 2dim(A). This does does NOT imply that the endomorphism algebra of a simple AV of CM type is a field, except in characteristic zero. Tate proved that ALL abelian varieties over the algebraic closure of a finite field are of CM type.

Shimura (one of the founders of the field!) defined an abelian variety to be of CM type its endomorphism algebra contains an algebraic number field of degree 2dim(A). This has the disadvantage that the product of two such abelian varieties need not be of the same type, but some authors follow him. [For example, take two elliptic curves whose endomorphism algebras are distinct CM fields. Then their product is not CM in Shimura's sense.]

There is also a disagreement: if an abelian variety becomes of CM type over an extension of the base field, some authors say it is of CM type, and some that it is potentially of CM type.