The geometry of some parameterizations and encodings

TL;DR

The paper investigates radical-based parameterizations of low genus algebraic curves over finite fields, demonstrating that many genus 2 curves can be parameterized by 3-radicals for large enough prime powers, enabling deterministic encodings.

Contribution

It proves that a positive proportion of genus 2 curves over large finite fields can be parameterized by 3-radicals, and extends the method to 5-radicals, providing explicit constructions.

Findings

A positive proportion of genus 2 curves are parameterizable by 3-radicals.

Deterministic encoding exists for certain finite fields when q ≡ 2 mod 3.

Explicit parameterizations are provided for 5-radicals.

Abstract

We explore parameterizations by radicals of low genera algebraic curves. We prove that for $q$ a prime power that is large enough and prime to $6$, a fixed positive proportion of all genus 2 curves over the field with $q$ elements can be parameterized by $3$-radicals. This results in the existence of a deterministic encoding into these curves when $q$ is congruent to $2$ modulo $3$. We extend this construction to parameterizations by $\ell$-radicals for small odd integers $\ell$, and make it explicit for $\ell=5$.