Canonical heights on the Jacobians of curves of genus 2 and the infinite descent

Abstract

Abstract. We give an algorithm to compute the canonical height on a Jacobian of a curve of genus 2. The computations involve only working with the Kummer surface and so lengthy computations with divisors in the Jacobian are avoided. We use this height algorithm to give an algorithm to perform the "infinite descent" stage of computing the Mordell-Weil group. This last stage is performed by a lattice enlarging procedure.

On the height constant for curves of genus two, II

“…It turns out that the bounds γ 2 ≤ 4 and γ 191 ≤ 2 given by [7] are sharp. At 941, however, we can improve on the bounds γ 941 ≤ 4 or γ 941 ≤ 2 (which is the improved bound from [7] or the bound from [3] that was obtained by looking at the Q 941 -rational points on K). Indeed, modulo 941, the polynomial (in homogeneous form) factors as Z(X −185Z) 5 .…”

“…In general, we can try to find γ exactly by searching through the points in K(k) that lift to J(k), in a similar way to the program mentioned in [3]. Table 1 tells us where to start off the search.…”

“…This set will contain all points of (canonical) height ≤ 2 (cf. [3] or [7]). By Proposition 7.1, these points, together with the P i , will generate the group we are looking for.…”

“…When γ is not too large, but 2 is, one can use the approach suggested by Flynn and Smart [3]. This consists in enumerating points up to (logarithmic) naive height γ + ε for a suitable ε > 0 and using the result to bound the index in the saturation.…”

“…Note that the method given in [3] for bounding the index of Λ in Λ can only bound the index by 4, no matter how far we go in the search of points in J(Q). Since in this case, it is known that the index must be odd, we still would have to exclude the possibility that the index is 3.…”