  We consider the issue of when the L-polynomial of one curve over $\F_q$
divides the L-polynomial of another curve. We prove a theorem which shows that
divisibility follows from a hypothesis that two curves have the same number of
points over infinitely many extensions of a certain type, and one other
assumption. We also present an application to a family of curves arising from a
conjecture about exponential sums. We make our own conjecture about
L-polynomials, and prove that this is equivalent to the exponential sums
conjecture.
