Maximal and minimal curves of the form $y^3=x^{(q^2+1)/2}+x$

Abstract

Let $p\ge 5$ be a prime with $p\equiv -1(\mathrm{mod}3)$, let $q=p^r$, and consider \[ \cC:\qquad y^3=x^{(q^2+1)/2}+x \] over $\F_{q^6}$. We prove the exact formula \[ \#\cC(\F_{q^6})=q^6+1+(-1)^{r+1}(q^2-1)q^3. \] Since $g(\cC)=(q^2-1)/2$, the curve is maximal when $r$ is odd and minimal when $r$ is even. The proof uses a birational Kummer model and an explicit Jacobi-sum point count. A congruence together with Frobenius invariance reduces the relevant Jacobi sums to cubic Gauss sums, whose sign is determined from the Fermat cubic. In particular, the maximality of $y^3=x^{13}+x$ over $\F_{5^6}$ appears as the first case of an infinite family.