On the height constant for curves of genus two, II

The Mordell–Weil sieve: proving non-existence of rational points on curves

“…See [25,27] for algorithms that provide these data in the case of genus 2 curves. From this information and the embedding C → J, we can then compute constants δ and d such thatĥ(ι(P )) dh(P ) + δ for all points P ∈ C(Q).…”

“…(with the same choice of projective coordinates for κ(P ) on both sides) or, in the notation of [27], t(P ) ⇐⇒ p (P ) = 0. …”

“…Note that in practice, the part of the computation that determines this Mordell-Weil basis can be rather time-consuming; but this is a different problem, and we will not consider it here. See [25][26][27] for the relevant algorithms. We also assume that we know a rational divisor of degree three on C. Again, it may not be easy to find such a divisor in practice.…”

Rational 6-Cycles Under Iteration of Quadratic Polynomials

“…For this, we would need to compute the canonical height of points in J(Q). For Jacobians of genus 2 curves, there is an explicit theory of heights (see [8,23,24]) that allows us to do that. For Jacobians of curves of genus 3, however, no comparable results seem to be currently available.…”

Integral points on hyperelliptic curves

“…To provably determine the MordellWeil group one however needs bounds for the difference between the logarithmic and canonical heights. For Jacobians of curves of genus 2 such bounds have been determined by Stoll [1999;2002], building on previous work of Flynn and Smart [1997]. At present, no such bounds have been determined for Jacobians of curves of genus ≥ 3, though work on this is in progress.…”