The probability that the number of points on the Jacobian of a genus 2 curve is prime

TL;DR

This paper generalizes heuristics for estimating the probability that the Jacobian of a genus 2 curve over a finite field has a prime number of points, extending previous work on elliptic curves and exploring related properties.

Contribution

It extends Galbraith and McKee's heuristics from elliptic curves to genus 2 Jacobians, analyzing prime point counts and asymptotic behaviors.

Findings

Heuristics for prime Jacobian point counts in genus 2

Probabilities of cyclicity and primality analyzed

Asymptotic behavior as genus increases discussed

Abstract

In 2000, Galbraith and McKee heuristically derived a formula that estimates the probability that a randomly chosen elliptic curve over a fixed finite prime field has a prime number of rational points. We show how their heuristics can be generalized to Jacobians of curves of higher genus. We then elaborate this in genus 2 and study various related issues, such as the probability of cyclicity and the probability of primality of the number of points on the curve itself. Finally, we discuss the asymptotic behavior as the genus tends to infinity.