The $a$-number, $p$-rank and Cartier points of genus 4 curves

TL;DR

This paper investigates genus 4 curves over finite fields, analyzing their Jacobian invariants, the existence of Cartier points, and providing statistical data for small primes to understand their structural properties.

Contribution

It offers the first comprehensive statistical analysis of genus 4 curves' invariants and Cartier points over finite fields, revealing bounds related to the $a$-number and $p$-rank.

Findings

Number of Cartier points is bounded for curves with $0 \\leq a < g$.

The number of Cartier points depends on the $a$-number and $p$-rank.

Collected data for primes 3, 5, 7, 11 to analyze invariants.

Abstract

We study genus $4$ curves over finite fields and two invariants of the $p$-torsion part of their Jacobians: the $a$-number ($a$) and $p$-rank ($f$). We collect and analyze statistical data of curves over $\mathbb{F}_p$ for $p=3,5,7,11$ and their invariants. Then, we study the existence of Cartier points, which are also related to the structure of $J[p]$. For curves with $0\leq a<g$, the number of Cartier points is bounded, and it depends on $a$ and $f$.