Deducing information about curves over finite fields from their Weil polynomials

TL;DR

This paper explores techniques to deduce properties of algebraic curves over finite fields from their Weil polynomials, aiding in curve classification and search strategies.

Contribution

It introduces new methods inspired by Serre's 1985 techniques to determine curve existence and properties from Weil polynomials.

Findings

Some Weil polynomials correspond to no curves, enabling their exclusion.

Methods can identify curves with specific automorphisms or maps to elliptic curves.

Techniques facilitate efficient search for curves with given Weil polynomials.

Abstract

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.