#6021: [with another new patch, needs review] Implement period lattices for
elliptic curves over CC
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Reporter: cremona | Owner: was
Type: enhancement | Status: new
Priority: major | Milestone: sage-4.0.1
Component: number theory | Keywords: elliptic curve periods
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Comment(by cremona):
Yet another patch, to be applied after the previous ones.
1. Better handling of precision. The algebraic quantities needed for
both periods and elliptic logs are now cached. Then period and log
computations just have to coerce into the appropriate Real/ComplexField,
and do the transcendental part via agm.
2. Elliptic log implementation now moved into period lattice class
(except for the algorithm="pari" case which is unchanged). Also available
via call i.e. as L.elliptic_logarithm(P) or just L(P). Uses an extended
agm function which has been separated off.
3. Earlier precision issues with a difficult example are fixed; we get
all the same digits as pari, and faster. To do this we compute the
extended AGM in double the required precision and then revert to desired
precision at the end. (I tried adding 10 or 20 bits of precision, but
that nasty example (18074g1) needs more).
The only remaining thing is to implement elliptic logs for non-real
lattices. This is not hard to do but harder to justify! Before I do
that, to test it I need to implement the reverse of the elliptic log --
using Weierstrass P-functions and derivative to go from z mod L back to
P(x,y) with complex coords in general.
Apologies for adding this after a positive review (which applies to the
first three patches only). So: the first 3 patches have a positive
review, while the 4th does not (yet).
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/6021#comment:17>
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