#4606: elliptic curves -- implement gross-Zagier L-functions
-------------------------+-------------------------------------------------
Reporter: | Owner:
robertwb | Status: needs_work
Type: | Milestone: sage-6.7
enhancement | Resolution:
Priority: major | Merged in:
Component: | Reviewers:
elliptic curves | Work issues:
Keywords: | Commit:
Authors: | c4daca81bcc855197d8196eb5caf20ddf7dea05a
Report Upstream: N/A | Stopgaps:
Branch: |
u/chapoton/4606 |
Dependencies: |
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Comment (by chapoton):
Hum, here is the current state of affairs:
{{{
sage: E = EllipticCurve('37a')
sage: K.<a> = QuadraticField(-40)
sage: A = K.class_group().gen(0)
sage: L = E.lseries_gross_zagier(A)
sage: LL = E.lseries_gross_zagier(A**2)
sage: L(2) + LL(2)
0.506799279512368
sage: E.lseries()(2) * E.quadratic_twist(-40).lseries()(2)
0.502803417587467
}}}
Not so good, in fact. Now let us compare Taylor expansions:
{{{
sage: L.taylor_series(2, 5)+LL.taylor_series(2, 5)
0.506799279512368 + 0.360199571567893*z - 0.122141848388581*z^2 -
0.00635398874570253*z^3 + 0.0383995215484257*z^4 + O(z^5)
sage: E.lseries().taylor_series(2,series_prec=5) *
E.quadratic_twist(-40).lseries().taylor_series(2,series_prec=5)
0.502803417587467 + 0.374948906665456*z - 0.144641137632262*z^2 +
0.00702138852027905*z^3 + 0.0487513598755609*z^4 + O(z^5)
}}}
Not far, but definitely not good. Note that the syntax of taylor expansion
differs.
--
Ticket URL: <http://trac.sagemath.org/ticket/4606#comment:23>
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