Fwd: [sage-nt] complex periods for elliptic curves

[William Stein]

Hi,

If somebody wanted to "assist" John Cremona with the project below,
that would be a good 580 project.

William


---------- Forwarded message ----------
From: John Cremona <john.c...@gmail.com>
Date: Mon, May 11, 2009 at 7:10 AM
Subject: [sage-nt] complex periods for elliptic curves
To: sag...@googlegroups.com



I'm going to implement an extension of the Period Lattice functions
for elliptic curves over number fields to handle the complex
embeddings.
Pari does not give these directly, but we can just use the complex AGM
to get a period basis.

1. The constructor will take an arbitrary embedding, not just a real
one.  The only way I know to tell if an embedding is real is to test
e(a).is_real() where e is the embedding and a is the generator.  Is
there a better way?

2. the field real_embedding will change its name to embedding;
together with a flag is_real.  I suggest using three values for that
flag: +1,0,-1 with +1 for real and Delta>0, -1 for real and Delta<0, 0
for non-real..

3. the methods is_rectangular() and real_period() will give a runtime
error for non-real embeddings.  Same for omega() -- though I could
make a case for omega() returning what is now complex_area for
non-real.

Does that sound reasonable?

After this will come the complex elliptic log, which is quite interesting...

John

PS the purpose of posting here instead of just opening a new ticket is
that if I just did that no-one here would see it.  I have opened a
ticket too: #6021





--
William Stein
Associate Professor of Mathematics
University of Washington
http://wstein.org