Isogeny Classes of Abelian Varieties over Function Fields

Let K be a field, K̄ its separable closure, Gal(K) = Gal(K̄/K) the (absolute) Galois group of K. Let X be an abelian variety over K. If n is a positive integer that is not divisible by char(K) then we write Xn for the kernel of multiplication by n in X(Ks). It is well-known [21] that Xn ia a free Z/nZ-module of rank 2dim(X); it is also a Galois submodule in X(K̄). We write K(Xn) for the field of definition of all points of order n; clearly, K(Xn)/K is a finite Galois extension, whose Galois group is canonically identified with the image of Gal(K) in Aut(Xn). We write IdX for the identity automorphism of X . We write End(X) for the ring of all K̄endomorphisms of X and End(X) for the corresponding Q-algebra End(X)⊗Q. We have Z · IdX ⊂ End(X) ⊂ End(X).