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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 thatdivisibility follows from a hypothesis that two curves have the same number ofpoints over infinitely many extensions of a certain type, and one otherassumption. We also present an application to a family of curves arising from aconjecture about exponential sums. We make our own conjecture aboutL-polynomials, and prove that this is equivalent to the exponential sumsconjecture.

Decomposing Jacobians of Curves over Finite Fields in the Absence of Algebraic Structure