  Suppose X is a (smooth projective irreducible algebraic) curve over a finite
field k. Counting the number of points on X over all finite field extensions of
k will not determine the curve uniquely. Actually, a famous theorem of Tate
implies that two such curves over k have the same zeta function (i.e., the same
number of points over all extensions of k) if and only if their corresponding
Jacobians are isogenous. We remedy this situation by showing that if, instead
of just the zeta function, all Dirichlet L-series of the two curves are equal
via an isomorphism of their Dirichlet character groups, then the curves are
isomorphic up to "Frobenius twists", i.e., up to automorphisms of the ground
field. Since L-series count points on a curve in a "weighted" way, we see that
weighted point counting determines a curve. In a sense, the result solves the
analogue of the isospectrality problem for curves over finite fields (also know
as the "arithmetic equivalence problem"): it says that a curve is determined by
"spectral" data, namely, eigenvalues of the Frobenius operator of k acting on
the cohomology groups of all l-adic sheaves corresponding to Dirichlet
characters. The method of proof is to shown that this is equivalent to the
respective class field theories of the curves being isomorphic as dynamical
systems, in a sense that we make precise.
