Monotonicity of maximal curve point counts in genus

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a prime power $q$ and integer $g\ge0$, set $N_q(g)=\max_C\#C(\mathbb F_q)$, where $C$ ranges over smooth projective geometrically connected genus-$g$ curves over $\mathbb F_q$. This is a finite maximum by the Hasse--Weil bound. "Increasing" here means nondecreasing; the same source's table allows equal adjacent values, so strict increase is not intended.

Statement recorded in the supplied source.

For every fixed prime power $q$, is $N_q(g+1)\ge N_q(g)$ for every integer $g\ge0$? [S2]

Short English statement

Can the largest possible number of rational points decrease when the genus increases by one?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8800050 (AMR-087-0050). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Everett W. Howe, Low-genus curves over finite fields: Problems and variations, IPAM Number Theory and Cryptography: Open Problems workshop (9--13 October 2006), PDF pages 39--41. https://helper.ipam.ucla.edu/publications/SCWS1/SCWS1_6237.pdf.