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Meta-ticket: add an implementation of Edixhoven's algorithm to Sage #12132

Open 1659f18b-8e7f-4ace-87e0-ea435f3ce618 opened 12 years ago

1659f18b-8e7f-4ace-87e0-ea435f3ce618 commented 12 years ago

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.

CC: @williamstein @pjbruin

Component: modular forms

Issue created by migration from https://trac.sagemath.org/ticket/12132

1659f18b-8e7f-4ace-87e0-ea435f3ce618 commented 12 years ago

Description changed:

--- 
+++ 
@@ -3,3 +3,21 @@
 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.
+
+Step 2: Improve the performance of finite field computations.
1659f18b-8e7f-4ace-87e0-ea435f3ce618 commented 12 years ago

Description changed:

--- 
+++ 
@@ -18,6 +18,6 @@

 For the mod p approach:

-Step 1: Implement finite commutative algebras over arbitrary fields.
+Step 1: Implement finite commutative algebras over arbitrary fields.  This is #12141.

 Step 2: Improve the performance of finite field computations.
1659f18b-8e7f-4ace-87e0-ea435f3ce618 commented 12 years ago

Description changed:

--- 
+++ 
@@ -20,4 +20,4 @@

 Step 1: Implement finite commutative algebras over arbitrary fields.  This is #12141.

-Step 2: Improve the performance of finite field computations.
+Step 2: Improve the performance of finite field computations.  This is #12142.
1659f18b-8e7f-4ace-87e0-ea435f3ce618 commented 12 years ago

Description changed:

--- 
+++ 
@@ -10,14 +10,12 @@

 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.
-
+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:

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