Abelian Varieties | Galois Representations and Properties of Ordinary Reduction

Introduction In this paper we will look at abelian varieties over number elds. We will be interested in particular in the number of places where such an abelian variety has ordinary reduction. Recall that if k is a eld of characteristic p and if X=k is an abelian variety, then X is said to be ordinary if Xp](k) = (Z=p) g , where g = dimX. If X is an abelian variety over any eld k, then, for each prime number`6 = p = char(k), the Galois group Gal(k sep =k) acts on the Tate module T ` X. For our purposes, the case where the eld k is nite will be of particular importance. It is well known that in this case the characteristic polynomial of the Frobenius element Fr 2 Gal(k=k) acting on T ` X has coeecients in Z and is independent of`. This means that for each`6 = p, the eigenvalues of Fr on T ` X are the same algebraic integers. The variety X is ordinary if and only if for some, or equivalently for any, valuation on Q extending the p-adic valuation on Q, precisely half these eigenvalues have valuation 0. Suppose that F is a number eld and that X=F is an abelian variety. At every nite place v of F, the residue eld F v is a nite eld. For all but nitely many of these places, the reduction X v =F v of X is an abelian variety. One can ask for how many valuations v this reduction is ordinary. From what we have seen above, it follows that the question whether X v is ordinary can be answered by looking at the eigenvalues of a Frobenius element Fr v 2 Gal(F=F) at v acting on T ` X = T ` X v , for any`with v(`) = 0. Note that the fact that X has good reduction at v implies that the image in End(T ` X) of such a Frobenius element is determined up to conjugation, and hence that the eigenvalues of this image are well deened. The only thing which seems to be known about this question in the general case can be found in Og, 2.7]. The result is that, after replacing F by a nite extension, there is a set P of places, of Dirichlet density 1, such that for each v 2 P, at …