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/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, Fℓ). 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 defined over F_p with complex multiplication by K such that the order #A(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.

Keywords: Complex multiplication, general symplectic group over a finite field, Jacobian over a finite prime field