  We discuss methods for using the Weil polynomial of an isogeny class of
abelian varieties over a finite field to determine properties of the curves (if
any) whose Jacobians lie in the isogeny class. Some methods are strong enough
to show that there are no curves with the given Weil polynomial, while other
methods can sometimes be used to show that a curve with the given Weil
polynomial must have nontrivial automorphisms, or must come provided with a map
of known degree to an elliptic curve with known trace. Such properties can
sometimes lead to efficient methods for searching for curves with the given
Weil polynomial.
  Many of the techniques we discuss were inspired by methods that Serre used in
his 1985 Harvard class on rational points on curves over finite fields. The
recent publication of the notes for this course gives an incentive for
reviewing the developments in the field that have occurred over the intervening
years.
