Surjectivity in Honda-Tate

We have seen that this map is well-defined and injective; we still have to show that it is surjective. In concrete terms, given a Weil q-integer π, we need to construct a simple abelian variety over Fq on which the Frobenius element acts as π. The only known construction is by reducing a CM abelian variety from characteristic zero. Specifically, our construction will proceed in five steps. First, fix a CM field L containing Q[π], and a CM type (L,Φ). (The field L and data Φ will be determined later.) Construct an abelian variety A0 over C of this CM type. In particular, A0 has an action by the element π ∈ L, and this automorphism has the desired characteristic polynomial. Second, show that A0 is in fact defined over a number field K. Third, show that A0 has good reduction at a place v of K over p. Thus the identity component of the special fiber is an abelian variety A, defined over some field of characteristic p, whose endomorphism ring contains L. Fourth, determine the Frobenius action on A. More precisely, we will see that the Frobenius element of A comes from an element πA of L, and determine its valuation at every place of L. By a judicious choice of (L,Φ) we can arrange that πA is “almost” π, in a sense which will be made precise. Fifth, argue that π itself must be in the image of the map HTq.