Extreme genus 3 curves over finite fields

Abstract: 

Curves over finite fields have a finite number of rational points, which number satisfies the Weil bounds. A basic question is whether the Weil bounds are sharp. It is known that they are for genus 1 and 2 curves --- genus 3 remains open.

Let k be a field with p^e elements, where e is even. Ibukiyama (1993) has shown that if e is 2 mod 4 then the upper bound is achieved by a genus 3 curve over k and if e is 0 mod 4 then the lower bound is achieved. If p is 3 mod 4 (and k is as above), I show there are genus 3 curves C and C’ over k satisfying the Weil upper and lower bounds. The construction shows that C and C’ are both hyperelliptic (and quadratic twists of each other). Ibukiyama’s construction used curves which were provably non-hyperelliptic.