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.