(An official version and a preprint are available online.)

We show that for every finite collection * C* of abelian varieties
over

Our explicit bound is expressed in terms of the Frobenius angles of
the elements of * C*. In general, suppose that

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 **F**_{2} is isogenous
to a product of elliptic curves over **F**_{2}, then the genus of *C* is
at most 26. This bound is sharp, because there is an
**F**_{2}-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 *J*_{0}(*N*) is isogenous over **Q**
to a product of elliptic curves.