Proceedings of the London Mathematical Society vol:104 issue:6 pages:1235-1270
In 2000, Galbraith and McKee heuristically derived a formula that estimates the probability that a randomly chosen elliptic curve over a xed nite prime eld 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 g = 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 for g --> 1.