This paper has now appeared in Acta Arith. 90, 183-201 (1999)
and so the preprint has been removed. Let *y*^{2} = 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 **P**^{3}), and denote by
*delta* : *K* --> *K*
the duplication map. For a point *P* in **P**^{3}(*k*)
with specified homogeneous coordinates
(*x*_{1} : *x*_{2} :
*x*_{3} : *x*_{4}),
denote
|*P*| = max{|*x*_{1}|, |*x*_{2}|,
|*x*_{3}|, |*x*_{4}|}
(where | . | is the absolute value on *k*). We derive a general bound
on the *height constant*
*c* = min_{P in K(k)}
|*delta*(*P*)|/|*P*|^{4},
namely
*c* >= |2^{4} 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.