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We discuss methods for using the Weil polynomial of an isogeny class ofabelian varieties over a finite field to determine properties of the curves (ifany) whose Jacobians lie in the isogeny class. Some methods are strong enoughto show that there are no curves with the given Weil polynomial, while othermethods can sometimes be used to show that a curve with the given Weilpolynomial must have nontrivial automorphisms, or must come provided with a mapof known degree to an elliptic curve with known trace. Such properties cansometimes lead to efficient methods for searching for curves with the givenWeil polynomial.  Many of the techniques we discuss were inspired by methods that Serre used inhis 1985 Harvard class on rational points on curves over finite fields. Therecent publication of the notes for this course gives an incentive forreviewing the developments in the field that have occurred over the interveningyears.

Deducing information about curves over finite fields from their Weil polynomials