Ariel Pacetti on Fri, 02 Oct 2009 00:16:53 +0200

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Heights over number fields

Dear Pari users,

is there a way to compute the height (even more, to compute the height matrix of points) for an elliptic curve over a number field?

In the case of rational curves, if one plugs in the minimal model, the routines work fine, but if one doesn't they do not. If one is working with an elliptic curve over a number field (say with non-trivial class group), then one cannot chose a minimal model. If I compute the height (or height matrix) representing the points in the number field with some precision, the routine gives some output. Is there a way to check this is the right answer?

This is a more general question than the application I have in mind. I just want to know is some points are linearly independent or not. I could compute the torsion of the given elliptic curve, and then try some linear combinations with coefficients inside a box (depending of the height of the points) and expect to find a linear dependence in the range if the points turn to be linearly dependent, but the height matrix kernel gives (without any effort) a certain answer in the rational case.