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Sage Days 18 Coding Sprint Projects

Contents

Sage Days 18 Coding Sprint Projects

Elliptic curves over function fields

This project will include the following topics:

Compute regulators of elliptic curves over function fields
Is there an algorithm to enumerate all elliptic curves over a function field of a given conductor?
Implement Tate's algorithm for elliptic curves over the function field F_p(t)
Verify BSD for elliptic curves over function fields of analytic rank 3 or higher
People: Sal Baig, William Stein, David Roe, Ken Ribet

Implement computation of the 3-Selmer rank of an elliptic curve over QQ

Some projects:

Implement the algorithm from Schaefer-Stoll which does the general case
Implement the algorithm in the reducible case due to Jeechul Woo, if we can find out what it is.
Compute the 3-Selmer ranks of all curves of conductor up to 1000
People: Robert Miller, William Stein, Victor Miller, Jeechul Woo (Noam's student) ??

Compute statistics about distribution of Heegner divisors and Kolyvagin divisors modulo primes p

Compute the reduction of x_1 using ternary quadratic forms, then use distribution relations and hit by Hecke operators to get reduction of all x_n. There is a theorem of Jetchev-Kane about the asymptotic distribution of x_n; compare our new data with that.
People: William Stein, Dimitar Jetchev, Drew Sutherland, Mirela Ciperiani, Ken Ribet, Victor Miller

Create a table of images of Galois representations, for elliptic curves

The goals of this project are:

Compute and record in a nice table the exact image of Galois in GL_2(F_p) for all p<60 and all curves in Cremona's tables, using Drew's new code/algorithm.
Extend the above to all p by using the explicit bound coded in Sage.
Extend the above to all p^k.
Compute the exact image for all curves of conductor up to 10^8 from the Stein-Watkins database. Add this data with some nice key to that database (i.e., change all the files to include a new field).
Think about images in GL(Z/mZ).
Think about statistics resulting from the above computation.

People: Drew Sutherland, Ken Ribet, William Stein

Fast computation of Heegner points

Implement the algorithm of Delauny/Watkins's algorithm for fast computation of Heegner points y_K \in E(K).
People: William Stein, Robert Bradshaw, Jen Balakrishnan

Implement code to compute the asymptotic distribution of Kolyvagin classes

This will be based on Jared Weinstein's talk.

People: Jared Weinstein, Mirela Ciperiani, William Stein

Verify Kolyvagin's conjecture for a specific rank 3 curve

This is done for examples of rank 2 curves. Nobody has ever checked that Kolyvagin's conjecture holds for a rank 3 curve.

Figure out exactly what needs to be computed and what might be an optimal curve and quadratic imaginary field to work with.
Verify that one Kolyvagin class for that curve is nonzero.
Possibly verify the conjecture for the first (known) rank 4 curve, which has conductor 234446. This would be computationally hard, but not impossible!

People: William Stein, Dimitar Jetchev, Victor Miller (sparse linear algebra), Jen Balakrishnan

Implement an algorithm in Sage to compute Stark-Heegner points

There is a new algorithm due to Darmon and Pollack for computing Stark Heegner point using overconvergent modular symbols. So this project would involve:

Implementing computation of overconvergent modular symbols.
Using an implementation of overconvergent modular symbols to implement the Stark-Heegner point algorithm.

People: Mathew Greenberg, Cameron Frank, Kiran Kedlay, Robert Pollack, Avner Ash

Compute the higher Heegner point y_5 on the curve 389a provably correctly

Implement an algorithm to compute the Gross-Zagier Rankin-Selberg convolution L-functions L(f,\chi,s), and use Zhang's formula to deduce heights of Heegner points.
Apply in the particular curves 389a for n=5.
Come up with an algorithm that is definitely right for provably computing Heegner points given the height.
Implement algorithm and run for 389a and n=5.
Make a table of heights of higher Heegner points. (Search to find any of height 0!)
Make a table of heights of derived Kolyvagin points.

People: Robert Bradshaw, William Stein, Jen Balakrishnan

Compute a Heegner point on the Jacobian of a genus 2 curve

People: Noam Elkies, virtually via his comments in this thread.

Visibility of Kolyvagin Classes

On the one hand, one of Kolyvagin's suite of conjectures is that the classes he constructs out of Heegner points ought to generate the entirety of Sha(E/K). On the other hand each element of Sha(E/K) is "visible" in some other abelian variety, in the sense of Mazur (see for instance http://www.williamstein.org/home/was/www/home/wstein/www/papers/visibility_of_sha/). I wonder if the Kolyvagin classes d(n) contributing to Sha become visible in abelian varieties in some *systematic* way; ie, in a modular Jacobian of possibly higher level. There are already some examples out there of computation with visibility and computation with Kolyvagin classes, so maybe we can gather some data.

People: Jared Weinstein, Mirela Ciperiani, William Stein, Dimitar Jetchev, Ken Ribet

Sage Tutorials

We would like to have a sequence of carefully prepared Sage tutorials on the following topics:

Introduction to Python (Kiran Kedlaya, 10am on Wednesday)
Tate's algorithm over number fields (David Roe, 10am on Thursday?)
2-descent in Sage (Robert Miller, 11am on Thursday)
Computing images of Galois representations (Drew Sutherland, William Stein, 11am on Friday)
Linear algebra modulo p (Robert Bradshaw, 11am on Wednesday)

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+   ← Revision 12 as of 2009-12-01 16:52:55 → ⇥
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-Deletions are marked like this.
+Additions are marked like this.
 Line 5:
-== Compute regulators of elliptic curves over function fields ==
+== Elliptic curves over function fields ==
 Line 7:
- People: ''Sal Baig''
+This project will include the following topics:
 Line 9:
-== Is there an algorithm to enumerate all elliptic curves over a function field of a given conductor? ==
+ * Compute regulators of elliptic curves over function fields
 Line 11:
- People: ''Sal Baig'', William Stein
+ * Is there an algorithm to enumerate all elliptic curves over a function field of a given conductor?
 Line 13:
+ * Implement Tate's algorithm for elliptic curves over the function field F_p(t)
-Line 14:
+Line 15:
-== Implement Tate's algorithm for elliptic curves over the function field F_p(t) ==
+ * Verify BSD for elliptic curves over function fields of analytic rank 3 or higher
-Line 16:
+Line 17:
- People: ''Sal Baig'', David Roe (?)

== Verify BSD for rank 3 or higher ==

 People: ''Sal Baig''
+ People: ''Sal Baig'', William Stein, David Roe, Ken Ribet
-Line 25:
+Line 21:
- People: ''Robert Miller'', William Stein
+Some projects:
  
 * Implement the algorithm from Schaefer-Stoll which does the general case

 * Implement the algorithm in the reducible case due to Jeechul Woo, if we can find out what it is.

 * Compute the 3-Selmer ranks of all curves of conductor up to 1000

 People: ''Robert Miller'', William Stein, Victor Miller, Jeechul Woo (Noam's student) ??
-Line 30:
+Line 34:
- People: ''William Stein'', Dimitar Jetchev
+ * Compute the reduction of $x_1$ using ternary quadratic forms, then use distribution relations and hit by Hecke operators to get reduction of all $x_n$.    There is a theorem of Jetchev-Kane about the asymptotic distribution of $x_n$; compare our new data with that. 

 People: ''William Stein'', Dimitar Jetchev, Drew Sutherland, Mirela Ciperiani, Ken Ribet, Victor Miller
-Line 33:
+Line 39:
-== Create a table of images of Galois representations, for elliptic curves and/or Jacobians, in some range ==
+== Create a table of images of Galois representations, for elliptic curves ==
-Line 35:
+Line 41:
- People: ''Drew Sutherland'', William Stein
+The goals of this project are:
-Line 37:
+Line 43:
-Drew Sutherland remarks: 
{{{
Hi William,
+  * Compute and record in a nice table the exact image of Galois in $GL_2(F_p)$ for all $p<60$ and all curves in Cremona's tables, using Drew's new code/algorithm.
-Line 41:
+Line 45:
-I would definitely be motivated to work on the table of Galois images project that you suggested in your list. I am currently rerunning my previous computations on the Stein-Watkins database using an improved version of the algorithm (just for the mod ell case at the moment, I still want to tweak the mod ell^k code some more). It would be great to get all this data organized and accessible in a useful form, especially while everything is fresh in my mind.
+  * Extend the above to all $p$ by using the explicit bound coded in Sage.
-Line 43:
+Line 47:
-Drew
}}}
+  * Extend the above to all $p^k$.

  * Compute the exact image for all curves of conductor up to $10^8$ from the Stein-Watkins database.  Add this data with some nice key to that database (i.e., change all the files to include a new field).

  * Think about images in $GL(Z/mZ)$. 

  * Think about statistics resulting from the above computation.
-Line 47:
+Line 56:
+ People: ''Drew Sutherland'', Ken Ribet, William Stein
-Line 48:
+Line 58:
-== Fully implement and optimize variant of Watkins's algorithm for fast computation of Heegner points ==
-Line 50:
+Line 59:
- People: ''William Stein'', Robert Bradshaw
+== Fast computation of Heegner points ==
-Line 52:
+Line 61:
-== Implement code to compute the asymptotic distribution of Kolyvagin classes (from Jared Weinstein's talk); this should be pretty easy, though generalizing to higher rank may be challenging ==
+ Implement the algorithm of Delauny/Watkins's algorithm for fast computation of Heegner points $y_K \in E(K)$.
-Line 54:
+Line 63:
- People: ''Jared Weinstein''
+ People: ''William Stein'', Robert Bradshaw, Jen Balakrishnan


== Implement code to compute the asymptotic distribution of Kolyvagin classes ==

This will be based on Jared Weinstein's talk.  

 People: ''Jared Weinstein'', Mirela Ciperiani, William Stein
-Line 58:
+Line 74:
- People: ''William Stein''
+This is done for examples of rank 2 curves.   Nobody has ever checked that Kolyvagin's conjecture holds for a rank 3 curve. 

  * Figure out exactly what needs to be computed and what might be an optimal curve and quadratic imaginary field to work with. 

  * Verify that one Kolyvagin class for that curve is nonzero. 

  * Possibly verify the conjecture for the first (known) rank 4 curve, which has conductor 234446.  This would be '''computationally hard''', but not impossible!

 People: ''William Stein'', Dimitar Jetchev, Victor Miller (sparse linear algebra), Jen Balakrishnan
-Line 61:
+Line 85:
+There is a new algorithm due to Darmon and Pollack for computing Stark Heegner point using overconvergent modular symbols.  So this project would involve:

  * Implementing computation of overconvergent modular symbols.
  * Using an implementation of overconvergent modular symbols to implement the Stark-Heegner point algorithm.
-Line 62:
+Line 91:
- People: ''Mathew Greenberg''
+ People: ''Mathew Greenberg'', Cameron Frank, Kiran Kedlay, Robert Pollack, Avner Ash
-Line 66:
+Line 96:
- People: ''Robert Bradshaw'', William Stein
+  * Implement an algorithm to compute the Gross-Zagier Rankin-Selberg convolution L-functions $L(f,\chi,s)$, and use Zhang's formula to deduce heights of Heegner points.
  * Apply in the particular curves 389a for n=5.
  * Come up with an algorithm that is definitely right for provably computing Heegner points given the height.
  * Implement algorithm and run for 389a and n=5.
  * Make a table of heights of higher Heegner points.   (Search to find any of height 0!)
  * Make a table of heights of derived Kolyvagin points.
-Line 68:
+Line 103:
-== Compute special values of the Gross-Zagier L-function L(f,chi,s)  ==
 
 People: ''Robert Bradshaw''

== Implement something toward the Pollack et al. overconvergent modular symbols algorithm ==

 People: ''Robert Pollack'', Avner Ash
+ People: ''Robert Bradshaw'', William Stein, Jen Balakrishnan
-Line 83:
+Line 111:
- People: ''Jared Weinstein'', William Stein
+On the one hand, one of Kolyvagin's suite of conjectures is that the classes he constructs out of Heegner points ought to generate the entirety of Sha(E/K).  On the other hand each element of Sha(E/K) is "visible" in some other abelian variety, in the sense of Mazur (see for instance http://www.williamstein.org/home/was/www/home/wstein/www/papers/visibility_of_sha/).  I wonder if the Kolyvagin classes d(n) contributing to Sha become visible in abelian varieties in some *systematic* way;  ie, in a modular Jacobian of possibly higher level.  There are already some examples out there of computation with visibility and computation with Kolyvagin classes, so maybe we can gather some data.
-Line 85:
+Line 113:
-This might be idle blather, but I've been thinking about Kolyvagin classes and I'm curious about the following.  On the one hand, one of Kolyvagin's suite of conjectures is that the classes he constructs out of Heegner points ought to generate the entirety of Sha(E/K).  On the other hand each element of Sha(E/K) is "visible" in some other abelian variety, in the sense of Mazur (see for instance http://www.williamstein.org/home/was/www/home/wstein/www/papers/visibility_of_sha/).  I wonder if the Kolyvagin classes d(n) contributing to Sha become visible in abelian varieties in some *systematic* way;  ie, in a modular Jacobian of possibly higher level.  There are already some examples out there of computation with visibility and computation with Kolyvagin classes, so maybe we can gather some data.
+ People: ''Jared Weinstein'', Mirela Ciperiani, William Stein, Dimitar Jetchev, Ken Ribet


== Sage Tutorials ==

We would like to have a sequence of carefully prepared Sage tutorials on the following topics:

   * Introduction to Python (Kiran Kedlaya, 10am on Wednesday)
   * Tate's algorithm over number fields (David Roe, 10am on Thursday?)
   * 2-descent in Sage (Robert Miller, 11am on Thursday)
   * Computing images of Galois representations (Drew Sutherland, William Stein, 11am on Friday)
   * Linear algebra modulo p (Robert Bradshaw, 11am on Wednesday)

Diff for "dayscambridge2/sprints"