Ute Spreckels (Oldenburg
19.05.2016 - W01 0-012 (Wechloy), 16 Uhr c.t.
On the Order of Abelian Varieties over Finite Prime Fields
Let A be a principally polarized CM abelian variety of dimension d defined over a number field F containing the CM-field K. If l is a prime number unramified in K, the Galois group G(l) of the l-division field of A lies in a maximal torus of the general symplectic group of dimension 2d over the finite field with l elements. Relying on a method of Weng, we explicitly write down this maximal torus as a matrix group. We restrict ourselves to the case that G(l) equals the maximal torus.
For P a prime ideal of F of inertia degree 1, let A(P) be the reduction of A modulo P. By counting matrices with eigenvalue 1 in G(l) we obtain a formula for the density of primes P such that l divides the order of A(P). Thereby we generalize results of Koblitz and Weng who computed this density for d=1 and 2. Both Koblitz and Weng also gave conjectural formulae for the number of primes P less than n such that A(P) has prime order.
We describe the involved heuristics and generalizations of these conjectures to arbitrary d. We provide examples with d=3.