Currently available as a preprint.

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.