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

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

2011-01-25

arxiv.org/abs/1101.4792v2

Mathematics

Number Theory

Minor edits, 37 pages. To appear in Proceedings of the London Mathematical Society

Scientific paper

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.

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