3,659 research outputs found
Computing canonical heights using arithmetic intersection theory
For several applications in the arithmetic of abelian varieties it is
important to compute canonical heights. Following Faltings and Hriljac, we show
how the canonical height on the Jacobian of a smooth projective curve can be
computed using arithmetic intersection theory on a regular model of the curve
in practice. In the case of hyperelliptic curves we present a complete
algorithm that has been implemented in Magma. Several examples are computed and
the behavior of the running time is discussed.Comment: 29 pages. Fixed typos and minor errors, restructured some sections.
Added new Example
Archimedean local height differences on elliptic curves
To compute generators for the Mordell-Weil group of an elliptic curve over a
number field, one needs to bound the difference between the naive and the
canonical height from above. We give an elementary and fast method to compute
an upper bound for the local contribution to this difference at an archimedean
place, which sometimes gives better results than previous algorithms.Comment: 10 pages, comments welcom
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