Canonical heights on genus-2 Jacobians (Q2520597)

Let \(C/\mathbb{Q}\) be a curve of genus 2, and let \(J\) be the Jacobian of \(C\). There is associated to \(J\) a canonical height function \(\hat{h}:J(\mathbb{Q})\to \mathbb{R}\) with the property that \(\hat{h}(nP)=n^2\hat{h}(P)\), and differing by at most a bounded amount from the naive Weil \(h\) height induced on \(J\) by the map to the Kummer surface \(J\to S\subseteq \mathbb{P}^3\). For various applications, such as finding generators for the Mordell-Weil group \(J(\mathbb{Q})\), one would like to be able to compute \(\hat{h}(P)\), and enumerate points \(P\) satisfying \(\hat{h}(P)\leq B\), for specified bound \(B\). There are known explicit bounds on the difference \(\hat{h}-h\), so in principle this is possible, but previous algorithms have been slow in practice. This paper introduces a practical polynomial-time algorithm addressing the first of these problems. In particular, the authors show how to compute \(\hat{h}(P)\) in time that is quasilinear in the size of the coefficients of \(C\) and coordinates of \(P\), and quasi-quadratic in the desired digits of precision. The authors also present improved methods for enumerating points of bounded canonical height, in practice making previously some infeasible searches for rational points manageable. Although the main result is stated over \(\mathbb{Q}\), most of the results hold in much greater generality, and the main result is expected to hold over an arbitrary number field. The main improvement to earlier efficient methods is to replace a factorization step, required to find primes of bad reduction, with a factorization into powers of coprimes due to \textit{D. J. Bernstein} [J. Algorithms 54, No. 1, 1--30 (2005; Zbl 1134.11352)].