ON THE ORDER OF ABELIAN SURFACES OF CM-TYPE OVER FINITE PRIME FIELDS

We consider a simple principally polarized abelian variety A of dimension g defined over a number field F with complex multiplication by an order in a CM-field K. Let l be a rational prime unramified in K/Q and let A[l] be the group of l-torsion points defined over the algebraic closure F-a. It is known that the Galois group Gal(F(A[l])/F) can be embedded into a maximal torus in the general symplectic group GSp(2g, F-l). We give an easy, explicit description of the maximal torus relating the splitting behaviour of l in K/Q to signed partitions of g. Applying our results to the case where A is an abelian surface, we are able to determine the density of primes p for which there exists an abelian variety (A) over bar defined over F-p with complex multiplication by K such that the order #(A) over bar (F-p) is divisible by l. We give a heuristic argument for the probability that the group of F-p-rational points on a simple, principally polarized abelian surface over F-p with complex multiplication has prime group order and present experimental data supporting our conjecture.