On the height constant for curves of genus two, by Michael Stoll

This paper has now appeared in Acta Arith. 90, 183-201 (1999) and so the preprint has been removed. Let y2 = f(x) (with f of degree 5 or 6) define a curve C of genus two over some non-archimedean local field k of characteristic different from 2. Suppose that f has integral coefficients. Let K be the Kummer surface associated to C (this is a surface in P3), and denote by delta : K --> K the duplication map. For a point P in P3(k) with specified homogeneous coordinates (x1 : x2 : x3 : x4), denote |P| = max{|x1|, |x2|, |x3|, |x4|} (where | . | is the absolute value on k). We derive a general bound on the  height constant   c = minP in K(k) |delta(P)|/|P|4, namely
c >= |24 disc(f)| ,
where disc(f) is the discriminant of f, viewed as a polynomial of degree 6.

Together with some bounds for the archimedean places, this gives an upper bound for the difference h - h^ between a certain naive height and the canonical height on the Jacobian of a curve of genus two defined over a number field. In particular, we get a bound on the naive height of a torsion point, which can be used to find the rational torsion subgroup of the Jacobian. We give a fairly detailed description of an algorithm that computes the rational torsion subgroup when Q is the base field.



Michael Stoll <stoll@math.uni-duesseldorf.de>