Asymptotic point-count conjecture for curves of fixed gonality

Faber–Grantham proposed that, for fixed $q$ and $\gamma\geq 2$, sufficiently high-genus curves of gonality $\gamma$ attain the universal point bound $\gamma(q+1)$. The supplied sources identify the all-$q$ statement as unresolved in characteristic $2$.

Known results

• The bound $\#C(\mathbb{F}_q)\leq\gamma(q+1)$ holds for curves of gonality $\gamma$.
• Gonality $\gamma=2$ satisfies $N_q(g,2)=2(q+1)$ for sufficiently large $g$, for every prime power $q$.
• For every fixed $q$ and $\gamma$, abelian-cover constructions give $\limsup_{g\to\infty}N_q(g,\gamma)=\gamma(q+1)$, but not eventual equality.

May 2023 odd-characteristic proof

On May 4, 2023, “Curves of fixed gonality with many rational points” reported that every odd prime power $q$ and every $\gamma\geq 2$ satisfy the conjecture for all sufficiently large genera. The result constructs nonsingular curves with exactly $\gamma(q+1)$ rational points; the supplied evidence does not establish the remaining even-$q$ cases.

Current status (as of September 2026): The conjecture is settled for $\gamma=2$ over all finite fields and is claimed for every $\gamma\geq 2$ over odd prime powers; the cases of even $q$ with $\gamma\geq 3$ remain open.