In this note we discuss the Hodge group Hdg(X) of a simple Abelian variety X of CM-type. It is well-known that dimℚHdg(X) ≤ dim(X). Assuming that X has somewhere good almost ordinary reduction, we prove that

dimℚ Hdg(X) = dim(X)

and give an explicit description of Hdg(X).

Almost ordinary Abelian varieties

Let A be an Abelian variety defined over a finite field k of characteristic p. We call A almost ordinary if dim(A) ≥ 1 and it has the same Newton polygon as the product of (dim(A) – 1)-dimensional ordinary Abelian variety and a supersingular elliptic curve. This means that its set of slopes is {0,1/2,1} and slope 1/2 has length 2. For example, an Abelian surface is almost ordinary if and only if it is neither ordinary nor supersingular. One may easily check that if g = dim (A) > 1 then A is almost ordinary if and only if its p-rank equals g – 1, i.e., the group of “physical” points of order p is isomorphic to (ℤ/pℤ)g–1.

Almost ordinary varieties were studied by Oort [13] in connection with the lifting problem of CM Abelian varieties to characteristic zero. In particular, he proved that each almost ordinary Abelian variety can be lifted to characteristic zero as CM Abelian variety (recall [26] that each Abelian variety over a finite field can be lifted to characteristic zero as CM Abelian variety up to an isogeny). Of course, if we start with an (absolutely) simple Abelian variety over a finite field, then its lifting will be also (absolutely) simple.