Let
be a number field
and let
be a curve of
genus
with Jacobian
variety . We study
the canonical height
.
More specifically, we consider the following two problems, which are important in
applications:
for a given
,
compute
efficiently;
for a given bound
,
find all
with
.
We develop an algorithm running in polynomial time (and fast in practice) to deal with
the first problem. For the second problem, we show how to tweak the naive height
to obtain significantly improved bounds for the difference
,
which allows a much faster enumeration of the desired set of points.
Our approach is to use the standard decomposition of
as a
sum of local “height correction functions”. We study these functions carefully, which
leads to efficient ways of computing them and to essentially optimal bounds. To get
our polynomial-time algorithm, we have to avoid the factorization step needed
to find the finite set of places where the correction might be nonzero. The
main innovation is to replace factorization into primes by factorization into
coprimes.
Most of our results are valid for more general fields with a set of absolute values
satisfying the product formula.
An errata was submitted on 30 Dec 2022 and posted onlineon 16 Feb 2023.
Keywords
canonical height, hyperelliptic curve, curve of genus 2,
Jacobian surface, Kummer surface
