Serre's genus 50 example

Abstract

This note presents explicit equations (up to birational equivalence over ${\mathbb{F}}_2$) for a complete, smooth, absolutely irreducible curve $X$ over ${\mathbb{F}}_2$ of genus $50$ satisfying $#X(\mathbb{F}_2)=40$. In his 1985 Harvard lecture notes on curves over finite fields, J-P.~Serre already showed the existence of such a curve: he used class field theory to describe the function field ${\mathbb{F}}_2(X)$ as a certain abelian extension of the function field ${\mathbb{F}}_2(E)$ of some elliptic curve $E/{\mathbb{F}}_2$. Although various more recent texts recall Serre's construction, explicit equations as well as a description of intermediate curves $X\to Y\to E$ over ${\mathbb{F}}_2$ seem to be new. We also describe explicit equations for a curve over ${\mathbb{F}}_2$ of genus $8$ with $11$ rational points, and for a curve over ${\mathbb{F}}_2$ of genus $22$ with $21$ rational points.

7 pages

References in corpus (1)

• Twists of Elliptic Curves