Opened 6 years ago

Last modified 4 years ago

#12132 new enhancement

Meta-ticket: add an implementation of Edixhoven's algorithm to Sage

Reported by: johanbosman Owned by: johanbosman
Priority: major Milestone: sage-wishlist
Component: modular forms Keywords:
Cc: was, pbruin Merged in:
Authors: Reviewers:
Report Upstream: N/A Work issues:
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Dependencies: Stopgaps:

Description (last modified by johanbosman)

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:

  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.
  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.
  3. We definitely also need LLL for matrices over QQ and RR. This is #12051.
  4. Jacobians, etc.

For the mod p approach:

  1. Implement finite commutative algebras over arbitrary fields. This is #12141.
  2. Improve the performance of finite field computations. This is #12142.

Change History (6)

comment:1 Changed 6 years ago by johanbosman

  • Cc was added

comment:2 Changed 6 years ago by johanbosman

  • Description modified (diff)
  • Milestone changed from sage-4.8 to sage-wishlist
  • Owner changed from craigcitro to johanbosman

comment:3 Changed 6 years ago by johanbosman

  • Description modified (diff)

comment:4 Changed 6 years ago by johanbosman

  • Description modified (diff)

comment:5 Changed 6 years ago by johanbosman

  • Description modified (diff)

comment:6 Changed 4 years ago by pbruin

  • Cc pbruin added
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