#21046: Numerical modular symbols for elliptic curves
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Reporter: wuthrich | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-7.3
Component: elliptic curves | Resolution:
Keywords: modular symbols | Merged in:
Authors: Chris Wuthrich | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
u/wuthrich/ticket/21046 | 63c6606552ea3a256b90596b609539853cb46755
Dependencies: #20864 | Stopgaps:
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Description changed by wuthrich:
Old description:
> I propose here to add fast modular symbols for elliptic curves. The
> proposed changes would add a cython file containing the new code to work
> with numerical modular symbols and integrate them for using for elliptic
> curves and their p-adic L-functions.
>
> The idea is similar to #6666, where "analytic modular symbols" were added
> to elliptic curves. However the code there is very slow and this ticket
> would replace that code completely.
>
> So a modular symbol for a given elliptic curve can be computed using
> numerical integration on the upper half plane rather than using linear
> algebra to determine the space of all modular symbols of level N first.
> Unlike #6666, we use rigorous bounds on the error of computations to be
> certain that we get the correct rational number.
>
> The code here compares in speed with eclib and is wayway faster than the
> python code within sage. When computing a single modular symbol or a few
> with small denominator, the code here is much faster than eclib and can
> cope with conductors in the millions. When computing all manin symbols
> for one curve, the speed is in the same order as for eclib for semistable
> curves, but sometimes slower.
>
> A link to the preprint explaining all this will be added here.
New description:
I propose here to add fast modular symbols for elliptic curves. The
proposed changes would add a cython file containing the new code to work
with numerical modular symbols and integrate them for using for elliptic
curves and their p-adic L-functions.
The idea is similar to #6666, where "analytic modular symbols" were added
to elliptic curves. However the code there is very slow and this ticket
would replace that code completely.
So a modular symbol for a given elliptic curve can be computed using
numerical integration on the upper half plane rather than using linear
algebra to determine the space of all modular symbols of level N first.
Unlike #6666, we use rigorous bounds on the error of computations to be
certain that we get the correct rational number.
The code here compares in speed with eclib and is wayway faster than the
python code within sage. When computing a single modular symbol or a few
with small denominator, the code here is much faster than eclib and can
cope with conductors in the millions. When computing all manin symbols for
one curve, the speed is in the same order as for eclib for semistable
curves, but sometimes slower.
The preprint explaining all is
[https://www.maths.nottingham.ac.uk/personal/cw/download/modsym.pdf here].
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Ticket URL: <https://trac.sagemath.org/ticket/21046#comment:2>
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