Zeta-functions of Curves over Finite Fields

TL;DR

This paper explores the construction of curves over finite fields with specific zeta-functions, analyzing conditions for their Jacobians to be isogenous to products of elliptic curves, with implications for cryptography and coding theory.

Contribution

It introduces methods to determine when a curve's zeta-function can be expressed as a product of elliptic curve zeta-functions and how to derive the curve's properties from this relationship.

Findings

Conditions for elliptic curves' zeta-functions to form a product representing a higher genus curve

Method to compute the characteristic polynomial of Frobenius for such curves

Potential applications in cryptography and coding theory

Abstract

Curves over finite fields are of great importance in cryptography and coding theory. Through studying their zeta-functions, we would be able to find out vital arithmetic and geometric information about them and their Jacobians, including the number of rational points on this kind of curves. In this paper, I investigate if it is possible to construct a curve over finite fields of a given genus $g$ whose zeta-function is given as a product of zeta-functions of $g$ elliptic curves, and find out alternative methods if it is not possible. Basically, I look for conditions which those $g$ elliptic curves should satisfy such that their product (of their Jacobians) is isogenous to the Jacobian of a curve of a given genus $g$. Then from this isogenous relationship I can determine the characteristic polynomial of the Frobenius endomorphism of the Jacobian of the new curve and by this characteristic polynomial I can thus determine the zeta-function of this new curve. By using the zeta-functions of curves in the form as generating functions, the number of rational points on curves can even be found out, which may lead to further researches relating to some applications in cryptography, coding theory and even information theory.