(An official version and a preprint are available online.)
We show that for every finite collection C of abelian varieties over Fq, there is an explicit bound B(C) such that every curve over Fq of genus greater than B(C) has a simple isogeny factor that does not occur in C.
Our explicit bound is expressed in terms of the Frobenius angles of the elements of C. In general, suppose that S is a finite nonempty collection of s real numbers in the interval [0, π]. If S = {0} set r = ½; otherwise, let r be twice the sum of ⎡(π/2) / θ⎤ over all nonzero θ∈ S. We show that if C is a curve over Fq whose genus is greater than either (q5s/2 + q-5s/2)/(2 cos(2π/5)) or (1 + √q)2r (1 + q-r) / 2 then C has a Frobenius angle θ such that neither θ nor -θ lies in S.
We do not claim that these genus bounds are sharp. For any particular set S we can usually obtain better bounds by solving a linear programming problem. For example, we show that if the Jacobian of a curve C over F2 is isogenous to a product of elliptic curves over F2, then the genus of C is at most 26. This bound is sharp, because there is an F2-rational model of the genus-26 modular curve X(11) whose Jacobian splits completely into elliptic curves.
As an application of our results, we give the complete list of integers N>0 such that the modular Jacobian J0(N) is isogenous over Q to a product of elliptic curves.