#9457: power series equality fails when trailing coefficients are zero
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Reporter: niles | Owner: malb
Type: defect | Status: needs_work
Priority: major | Milestone: sage-6.1
Component: commutative | Resolution:
algebra | Merged in:
Keywords: | Reviewers:
Authors: niles | Work issues: resolve problems with
Report Upstream: N/A | sha_tate.py
Branch: | Commit:
u/niles/ticket/9457 | e2219602a9d360f465ad79b098e4f923765ab791
Dependencies: | Stopgaps: #15748
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Comment (by niles):
Replying to [comment:26 wuthrich]:
> Of course, I offer my help with this as I am at the origin of the code
that causes troubles.
Thanks :) The last real progress I made in understanding this is up in
comments [comment:13] and [comment:14], where I worked through the
offending doctest by hand -- see
[https://github.com/sagemath/sage/blob/master/src/sage/schemes/elliptic_curves/sha_tate.py#L632
the block starting on line 632].
One helpful thing would be to verify the calculations below:
Using the objects constructed in comment [comment:13], sage says
{{{
sage: G
O(5^3) + (2*5^-1 + O(5^0))*T + O(T^2)
sage: H
O(T^2)
sage: lpv
(O(5^3) + (2*5^-1 + O(5^0))*T + O(T^2), O(T^2))
sage: eps.transpose()
[ 5/9 25/18]
[-5/18 5/9]
}}}
I think that `H` is not correct, as it it built from the following list of
coefficients
{{{
sage: [lps[n][1] for n in range(0,lps.prec())]
[O(5^2), O(5^-1), O(5^-1), O(5^-1), O(5^-1)]
}}}
So I think the correct value should be `H = O(5^2) + O(5^-1)*T + O(T^2)`
and hence
`-R(p) * H = O(5^3) + O(5^0)*T + O(T^2)`. Converting `eps.transpose()` to
the 5-adics `R`, so that I can check `lpv * eps.transpose()` by hand, I
have
{{{
sage: Matrix(R,2,2,[R(_) for _ in eps.transpose().list()])
[ 4*5 + 2*5^2 + 5^4 + 2*5^5 + O(5^6) 2*5^2 + 5^3 + 3*5^5 + 3*5^6
+ O(5^7)]
[3*5 + 3*5^2 + 4*5^3 + 5^4 + 5^5 + O(5^6) 4*5 + 2*5^2 + 5^4 + 2*5^5
+ O(5^6)]
}}}
Therefore `lpv * eps.transpose()` should be
{{{
( O(5^3) + (3 + O(5))*T + O(T^2) , O(5^4) + O(5^1)*T + O(T^2) )
}}}
Can you verify these calculations, and verify that if this were the output
of `lpv*eps.transpose()`, then the rest of the code would work correctly
and return `1`, as expected? (I think I was confused about this last
point when I made my comment [comment:14]; but this value for `lpv *
eps.transpose()` should make `shan0 = 1` and `shan1 = 0`, and if I read
the rest of that block correctly then this will lead to the expected
output).
----------
If the analysis above is right, then the only problem remaining to solve
is why `H` is not computed correctly. The problem seems to be that the
power series ring over p-adics does not keep track of negative precision
when there are no non-zero coefficients:
{{{
sage: R(0).add_bigoh(-1)
O(5^-1)
sage: QpT(R(0).add_bigoh(-1))
0 # should be same as line above
sage: QpT(1/25 + R(0).add_bigoh(-1))
5^-2 + O(5^-1) # precision remembered when non-zero coefficients are
present
}}}
I guess there is some problem with positive precision too:
{{{
sage: QpT(R(0).add_bigoh(1))
0
sage: QpT(1 + R(0).add_bigoh(1))
1 + O(5)
}}}
Is this issue already part of one of the other padic bug tickets?
--
Ticket URL: <http://trac.sagemath.org/ticket/9457#comment:27>
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