On the order of abelian surfaces of cm-type over finite prime fields

Abstract

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/ 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, ). We give an easy, explicit description of the maximal torus relating the splitting behaviour of ℓ in K/ 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 defined over with complex multiplication by K such that the order #() is divisible by ℓ. We give a heuristic argument for the probability that the group of rational points on a simple, principally polarized abelian surface over with complex multiplication has prime group order and present experimental data supporting our conjecture.