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The geometry of some parameterizations and encodings

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  • 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 $l$-radicals for small odd integers $l$, and make it explicit for $l=5$.
    Mathematics Subject Classification: Primary: 14H45, 14G50, 14E20; Secondary: 11G20, 12G05, 11S20.


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