Canonical Heights on Genus Two Jacobians
J. Steffen M\"uller, Michael Stoll

TL;DR
This paper develops efficient algorithms for computing the canonical height on genus two Jacobians over number fields, improving bounds for height differences and enabling faster enumeration of points with bounded height.
Contribution
It introduces a polynomial-time algorithm for computing canonical heights on genus two Jacobians and refines height bounds to accelerate point enumeration.
Findings
Algorithm runs in polynomial time and is practical.
Improved bounds for height differences enable faster point enumeration.
Replaces prime factorization with coprime factorization for efficiency.
Abstract
Let be a number field and let be a curve of genus 2 with Jacobian variety . In this paper, we study the canonical height . More specifically, we consider the following two problems, which are important in applications: (1) for a given , compute efficiently; (2) for a given bound , find all with . We develop an algorithm running in polynomial time (and fast in practice) to deal with the first problem. Regarding the second problem, we show how one can tweak the naive height that is usually used to obtain significantly improved bounds for the difference , which allows a much faster enumeration of the desired set of points. Our approach is to use the standard decomposition of as a sum of local `height correction functions'. We study…
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