#6021: [with new patches, 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):
OK, so I have added two new patches: embed_qqbar.patch is independent of
the others, and adds functionality to the refine_embedding() function for
embeddings of number fields to allow extending any embedding into RR or CC
to an embedding into AA or QQbar (choosing the correct root). Then
cperiods.patch uses that to rethink how precision is handled for period
lattices (real and complex): the embedding supplied by the user is only
used to determine which embedding is wanted, not its precision (which is
ignored). the lattice converts this (on construction) into an embedding
into AA or QQbar, which it keeps. Then the period-finding functions do as
much as they can exactly (i.e. in AA or QQbar), up as far as computing
sqrt(ei-ej) where e1,e2,e3 are the roots of the 2-division polynomial (so
these expressions may have degree 6* the degree of the field); only then
are they converted into RR or CC (with the correct precision) for the
transcendental step of computing AGMs.
To test this I removed the "..." which worried robertwb and reran the
doctests -- they now all pass and agree on both 32 and 64-bit machines.
I think this is a good strategy to use for number field embeddings (a.k.a.
infinite places): separate the information "which place" from the
information "what precision".
If this is approved, I will use the same tactic to improve the precision
of the elliptic log (since the next thing on this agenda is to implement
complex elliptic logs).
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/6021#comment:14>
Sage <http://sagemath.org/>
Sage - Open Source Mathematical Software: Building the Car Instead of
Reinventing the Wheel
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