#12132: Meta-ticket: add an implementation of Edixhoven's algorithm to Sage
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Reporter: johanbosman | Owner: johanbosman
Type: enhancement | Status: new
Priority: major | Milestone: sage-wishlist
Component: modular forms | Keywords:
Work_issues: | Upstream: N/A
Reviewer: | Author:
Merged: | Dependencies:
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Description changed by johanbosman:
Old description:
> A book on the computational aspects of Galois representations associated
> with modular forms was published last summer: http://www.math.univ-
> toulouse.fr/~couveig/publi/book.pdf
>
> At http://www.sagenb.org/home/pub/3154/ one can find a talk I gave
> containing a Sage implementation of this algorithm.
>
> The purpose of this ticket is to polish up and move that implementation
> to Sage.
>
> There are in fact two approaches. Both are based on computing with
> torsion point of Jacobians of modular curves. The first approach goes
> via complex approximations and has already led to some practical results.
> The second approach is via modulo p computations; this works very well in
> theory, but practical implementations have not been well worked out yet.
>
> Let us start by working out the first few steps in each of these
> approaches.
>
> For the numerical approach:
>
> Step 1: Implement the upper half plane. There already seems to be a
> ticket for this (#9439), but the patch given there needs a vast
> improvement. In fact, what we need is completely disjoint from what is
> done there.
>
> Step 2: Implement numerical evaluation of modular forms at upper half
> plane points and numerical integration of modular forms between points in
> the (extended) upper half plane.
>
> Step 3: Jacobians, etc.
>
> For the mod p approach:
>
> Step 1: Implement finite commutative algebras over arbitrary fields.
> This is #12141.
>
> Step 2: Improve the performance of finite field computations.
New description:
A book on the computational aspects of Galois representations associated
with modular forms was published last summer: http://www.math.univ-
toulouse.fr/~couveig/publi/book.pdf
At http://www.sagenb.org/home/pub/3154/ one can find a talk I gave
containing a Sage implementation of this algorithm.
The purpose of this ticket is to polish up and move that implementation to
Sage.
There are in fact two approaches. Both are based on computing with
torsion point of Jacobians of modular curves. The first approach goes via
complex approximations and has already led to some practical results. The
second approach is via modulo p computations; this works very well in
theory, but practical implementations have not been well worked out yet.
Let us start by working out the first few steps in each of these
approaches.
For the numerical approach:
Step 1: Implement the upper half plane. There already seems to be a
ticket for this (#9439), but the patch given there needs a vast
improvement. In fact, what we need is completely disjoint from what is
done there.
Step 2: Implement numerical evaluation of modular forms at upper half
plane points and numerical integration of modular forms between points in
the (extended) upper half plane.
Step 3: Jacobians, etc.
For the mod p approach:
Step 1: Implement finite commutative algebras over arbitrary fields. This
is #12141.
Step 2: Improve the performance of finite field computations. This is
#12142.
--
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/12132#comment:4>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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