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

Wouter Castryck, Amanda Folsom, Hendrik Hubrechts, Andrew V. Sutherland

Published 2011-01-25, updated 2011-09-30Version 2

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.