9 problems

Third-largest-genus conjecture. The candidate for the third largest genus is the maximal curve of genus

Let $q$ be a prime power, and let $(*_2)$ denote the second case considered in the paper for an $\mathbf F_{q^2}$-maximal curve. A plane $\mathbf F_{q^2}$-model is a plane equation…

Let $q$ be a prime power, and let $c_0(q+1,r)$ and $c_1(q+1,r)$ denote the Castelnuovo and Halphen numbers, respectively, in the notation of the paper. An $\mathbf F_{q^2}$-maximal…

Fukasawa–García–Torres plane-model conjecture. The curve $X$ is $\mathbb F_{\ell^2}$-isomorphic to the above plane model with ${\rm deg}(F)=m$.

Classification conjecture. The genus satisfies

Necessary-condition conjecture. This embedding property is a necessary condition for a maximal curve to be covered by the Hermitian curve.

Let $X$ be a maximal curve over $k$ of genus $g>0$. Assume that there is a rational point $P_0\in X(k)$ whose first non-gap is $m_1$, and let $n+1$ be the dimension of the complete…

Maximality divisibility conjecture. If $\overline{C}_{R,r}$ is $\mathbb{F}_{p^{2f}}$-maximal, then

Hermitian covering conjecture. Every $\mathbb{F}$-maximal curve is covered by a Hermitian curve.