This talk is based on joint work with Noam Elkies and
Christophe Ritzenthaler.

Suppose you are given a finite set S of simple abelian varieties
over a finite field k. Is there a bound on the genera of the
curves over k whose Jacobians are isogenous to products
of powers of elements of S?

Serre, using results of Tsfasman and Vladuts, showed that the
answer is yes. We give explicit bounds on the genus, in terms
of the "Frobenius eigenvalues" (the roots of the characteristic
polynomials of Frobenius) of the elements of S.

We show, for example, that the maximal genus of a curve over
F_2 whose Jacobian splits completely (up to isogeny) into
a product of elliptic curves is 26 --- a bound that is
attained by a certain model of the modular curve X(11).