#10748: Extend p-adic L-series to handle nontrivial Teichmuller components
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Reporter: davidloeffler | Owner: cremona
Type: enhancement | Status: positive_review
Priority: major | Milestone: sage-4.7
Component: elliptic curves | Keywords: p-adic L-function
Author: David Loeffler | Upstream: N/A
Reviewer: Chris Wuthrich | Merged:
Work_issues: |
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Changes (by wuthrich):
* status: needs_review => positive_review
* reviewer: => Chris Wuthrich
Comment:
All tests passed.
Here is a further example to illustrate the code. Maybe, we should add it
in the docstring somewhere ?
{{{
sage: E=EllipticCurve('11a1')
sage: lp=E.padic_lseries(7)
sage: lp.series(4,eta=1)
6 + 2*7^3 + 5*7^4 + O(7^6) + (4*7 + 2*7^2 + O(7^3))*T + (2 + 3*7^2 +
O(7^3))*T^2 + (1 + 2*7 + 2*7^2 + O(7^3))*T^3 + (1 + 3*7^2 + O(7^3))*T^4 +
O(T^5)
sage: lp.series(4,eta=2)
5 + 6*7 + 4*7^2 + 2*7^3 + 3*7^4 + 2*7^5 + O(7^6) + (6 + 4*7 + 7^2 +
O(7^3))*T + (3 + 2*7^2 + O(7^3))*T^2 + (1 + 4*7 + 7^2 + O(7^3))*T^3 + (6 +
6*7 + 6*7^2 + O(7^3))*T^4 + O(T^5)
sage: lp.series(4,eta=3)
O(7^6) + (3 + 2*7 + 5*7^2 + O(7^3))*T + (5 + 4*7 + 5*7^2 + O(7^3))*T^2 +
(3*7 + 7^2 + O(7^3))*T^3 + (2*7 + 7^2 + O(7^3))*T^4 + O(T^5)
}}}
The last vanishes at T=0, which is good, because it corresponds to the
positive rank over sqrt(-7):
{{{
sage: E.quadratic_twist(-7).rank()
1
}}}
This proves that E has rank 1 over Q(zeta_7).
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/10748#comment:4>
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