American Institute of Mathematical Sciences

November  2014, 8(4): 437-458. doi: 10.3934/amc.2014.8.437

The geometry of some parameterizations and encodings

 1 Université de Bordeaux, IMB, UMR 5251, F-33400 Talence, France 2 DGA.MI, La Roche Marguerite, F-35174 Bruz, France

Received  May 2014 Revised  June 2014 Published  November 2014

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 $l=5$.
Citation: Jean-Marc Couveignes, Reynald Lercier. The geometry of some parameterizations and encodings. Advances in Mathematics of Communications, 2014, 8 (4) : 437-458. doi: 10.3934/amc.2014.8.437
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