From the introduction: Let \(C\) be a curve of genus two over \(\mathbb Q\), and let \(J\) denote its Jacobian. We choose a point \(P\in J(\mathbb Q)\) and want to compute its canonical height with Flynn and Smart’s algorithm [E. V. Flynn and N. P. Smart, Acta Arith. 79, 333–352 (1997; Zbl 0895.11026)]. Its first step consists in finding a so-called “good multiple” \(nP\) of the given point, by computing \(2P\), \(3P\), and so on, until we find an \(n\) such that \(\varepsilon_p(nP)=0\) for all finite primes \(p\). Here, \(\varepsilon_p(P)\) is defined as follows. Let \(x= (x_1,x_2,x_3,x_4)\) be projective coordinates for the image of \(P\in J(\mathbb Q)\) on the Kummer surface \(K\subset\mathbb P^3\). Then (with the notation \(v_p(x)= \min\{v_p(x_1),\dots, v_p(x_4)\}\))
\[ \varepsilon_p(P)= v_p(\delta(x))- 4v_p(x), \] where \(v_p\) is the additive \(p\)-adic valuation (such that \(v_p(p)=1\)) and where \(\delta\) is the duplication map on \(K\).
In this paper, we investigate the functions \(\varepsilon_p\) more closely. For this purpose, we replace our base field \(\mathbb Q\) by some \(p\)-adic field (or, more generally, a non-archimedean local field of characteristic different from 2) \(k\) with additive valuation \(v\). We define the function \(\varepsilon\) on \(J(k)\) as above by \[ \varepsilon(P)= v(\delta(x))- 4v(x), \] where \(x= (x_1,x_2,x_3,x_4)\) is some set of Kummer coordinates of \(P\) (meaning projective coordinates for the image of \(P\) on \(K\)). Our main result is the following.
Theorem 1.1. Let \(U= \{P\in J(k)\mid \varepsilon(P)=0\}\). Then \(U\) is a subgroup of finite index in \(J(k)\), and \(\varepsilon(P)\) depends only on the coset of \(P\bmod U\).
The first part proves that Flynn and Smart’s algorithm is correct. Both statements together lead to an improved algorithm for the height computation.
In the first paper of this series [M. Stoll, Acta Arith. 90, No. 2, 183–201 (1999; Zbl 0932.11043)], we used representation theory obtain general bounds on the height constant
\[ \gamma= \max_{P\in J(k)} \varepsilon(P), \]
in terms of the discriminant of the curve. As a by-product of the results derived in the present paper, we can get considerable improvements in bounding \(\gamma\). This can be applied in order to find generators of the Mordell-Weil group of the Jacobian of a genus two curve over \(\mathbb Q\). We discuss this in some detail and give two examples (of ranks 7 and 12, respectively) to demonstrate the method.
It should be noted that similar results hold for elliptic curves. See for example S. Siksek [Rocky Mt. J. Math. 25, 1501–1538 (1995; Zbl 0852.11028)], where this approach is used to get good bounds for the height constant.