#4741: [with patch, not yet ready for review] Implement S-integral point finding
for elliptic curves over Q
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Reporter: cremona | Owner: was
Type: enhancement | Status: new
Priority: major | Milestone: sage-3.3
Component: number theory | Resolution:
Keywords: |
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Comment (by cremona):
The second patch fixes various issues with the padic_log function:
1. Bug: points of finite order have log 0 and this case should be
dealt with at the beginning. The code breaks on E(0), for example.
2. I deleted the "print_mode" parameter since it was actually changing
the output type from p-adic to rational; callers can lift if they want.
3. The test "if xde == p**(xde.valuation(p))" seems to be wrong. I
changed it to "if x.valuation(p)>=0", which is the condition that P is not
in the formal group.
4. It did not multiply P by the Tamagawa exponent.
5. It did not divide the answer by the factor you multiplied the point
by!
6. The precision was artificially capped at 20, since the prec
parameter to log() was not used and defaults to 20. That is the precision
of the power series in t, where we will substitute t=-x/y, so I added
"prec=1+precision//v" where v is the valuation of -x/y. (N.B. in
{{{E^1}}} one has val(x)=-2*v and val(y)=-3*v for some v>0, so
v=val(-x/y).
7. I added some doctests to show that log(k*P)/log(P)==k in various
examples. That is how I discovered the precision problem. Now it looks
pretty good.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/4741#comment:1>
Sage <http://sagemath.org/>
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