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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:
- This worksheet does it using Tate's suggestion (i.e., use the definition): http://sagenb.org/home/pub/1198/
Is there an algorithm to enumerate all elliptic curves over a function field of a given conductor?
- Victor Miller reports that there are some papers that give efficient algorithms for solving "S-unit equations" over function fields, which seems relevant.
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
Robert Bradshaw implemented faster arithmetic for F_p(t): #7585

People: Sal Baig, William Stein, David Roe, Ken Ribet, Kiran Kedlaya, Robert Bradshaw, Victor Miller (S-unit equations), Thomas Barnet-Lamb

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: Jeechul Woo's GP script
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; around only Thu, Fri)

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.
Stein: I posted a bundle based against Sage-4.2.1 here (called heegner-4.2.1.hg), which has highly relevant code: http://trac.sagemath.org/sage_trac/ticket/6616

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, Kiran Kedlaya, David Roe

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. See http://wstein.org/misc/sagedays18_papers/weinstein-kolyvagin_classes_for_higher_rank_elliptic_curves.pdf

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: Some details for 5077a
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, Robert Bradshaw

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: Matthew Greenberg, Cameron Frank, Kiran Kedlaya, Robert Pollack, Avner Ash, David Roe, Jay Pottharst, Thomas Barnet-Lamb

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

Jared describes the problem thus: 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.
I wrote a short paper with Dimitar Jetchev in 2005 (which I forgot about until just now, and never published), which I think was motivated by this question: :jetchev-stein-congruences_and_unramified_cohomology.pdf

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

Find an algorithm to decide if y_{p^n} is divisible by (g-1) and run it for a curve of rank >= 2

Consider the Heegner y_{p^n} over the anticyclotomic tower for a rank >= 2 curve, Sha trivial, etc.
Barry Mazur remarks that this may be connected to his notion (with Rubin) of "Shadow lines" in Mordell-Weil.

People: Mirela Ciperiani, William Stein, Barry Mazur, Jay Pottharst

Compute Frobenius eigenvalues for a bunch of curves to illustrate Katz-Sarnak

curves that vary in various ways, e.g., coverings of one curve...
might use David Harvey's super fast Kedlaya zeta function implementation in Sage (his C++ code), but unfortunately the curves that typically arise aren't hyperelliptic, and Harvey's code only applies to hyperelliptic curves.
could lead to questions of an "anabelian sort"... e.g., take elliptic curve over finite field, take n-torsion points, extract roots, get coverings, etc. Compute Frobenius eigenvalues of all these covers.
To do a useful computation, one needs to compute Frobenius, say for the curve y^{\ell} = f_N(x), where f_N(x) is the N-th division polynomial on an elliptic curve. This following paper is about how to count points on superelliptic curves: Gaudry-Gurel
There is a better paper maybe about this super-elliptic algorithm, with a Magma implementation here: http://www.math.tu-berlin.de/~minzlaff/

People: Barry Mazur, Kiran Kedlaya, Thomas Barnet-Lamb, David Harvey, Mirela Ciperiani, Sal Baig (lots of possibly relevant data over function fields)

Sage Tutorials

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

Introduction to Python/Sage (Kiran Kedlaya, 10am on Wednesday)
Linear algebra modulo p (Robert Bradshaw, 11am 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 and William Stein, 11am on Friday)

-  ⇤ ← Revision 8 as of 2009-11-29 05:21:43 → 
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+   ← Revision 27 as of 2009-12-03 02:55:21 → ⇥
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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: 
    * This worksheet does it using Tate's suggestion (i.e., use the definition): http://sagenb.org/home/pub/1198/
-Line 11:
+Line 12:
- People: ''Sal Baig'', William Stein
+ * Is there an algorithm to enumerate all elliptic curves over a function field of a given conductor?   
    * Victor Miller reports that there are some papers that give efficient algorithms for solving "S-unit equations" over function fields, which seems relevant.
-Line 13:
+Line 15:
+ * Implement Tate's algorithm for elliptic curves over the function field F_p(t)
-Line 14:
+Line 17:
-== 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 19:
- People: ''Sal Baig'', David Roe (?)
+ * Robert Bradshaw implemented faster arithmetic for $F_p(t)$: [[http://trac.sagemath.org/sage_trac/ticket/7585|#7585]]

 

People: ''Sal Baig'', William Stein, David Roe, Ken Ribet, Kiran Kedlaya, Robert Bradshaw, Victor Miller (S-unit equations), Thomas Barnet-Lamb
-Line 21:
+Line 28:
- 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: [[attachment:ThreeDescent.gp|Jeechul Woo's GP script]]

 * 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; around only Thu, Fri)
-Line 26:
+Line 41:
- 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. 

 * Stein: I posted a bundle based against Sage-4.2.1 here (called heegner-4.2.1.hg), which has highly relevant code: http://trac.sagemath.org/sage_trac/ticket/6616

People: ''William Stein'', Dimitar Jetchev, Drew Sutherland, Mirela Ciperiani, Ken Ribet, Victor Miller
-Line 29:
+Line 48:
-== 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 31:
+Line 50:
- People: ''Drew Sutherland'', William Stein
+The goals of this project are:
-Line 33:
+Line 52:
-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 37:
+Line 54:
-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 39:
+Line 56:
-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 43:
+Line 65:
+People: ''Drew Sutherland'', Ken Ribet, William Stein, Kiran Kedlaya, David Roe
-Line 44:
+Line 67:
-== Fully implement and optimize variant of Watkins's algorithm for fast computation of Heegner points ==
-Line 46:
+Line 68:
- People: ''William Stein'', Robert Bradshaw
+== Fast computation of Heegner points ==
-Line 48:
+Line 70:
-== 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 50:
+Line 72:
- 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.   See http://wstein.org/misc/sagedays18_papers/weinstein-kolyvagin_classes_for_higher_rank_elliptic_curves.pdf

People: ''Jared Weinstein'', Mirela Ciperiani, William Stein
-Line 54:
+Line 83:
- 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:  [[/5077aestimates|Some details for 5077a]]

  * 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, Robert Bradshaw
-Line 57:
+Line 94:
+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 58:
+Line 100:
- People: ''Mathew Greenberg''
+People: ''Matthew Greenberg'', Cameron Frank, Kiran Kedlaya, Robert Pollack, Avner Ash, David Roe, Jay Pottharst, Thomas Barnet-Lamb
-Line 62:
+Line 104:
- 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 64:
+Line 111:
-== 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 75:
+Line 115:
- People: Noam Elkies, virtually via his [[http://groups.google.com/group/sageday18/browse_thread/thread/6904bf31c59bb444|comments in this thread]].
+People: Noam Elkies, virtually via his [[http://groups.google.com/group/sageday18/browse_thread/thread/6904bf31c59bb444|comments in this thread]].
-Line 79:
+Line 119:
- People: ''Jared Weinstein'', William Stein
+  * Jared describes the problem thus:  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 81:
+Line 121:
-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.
+  * I wrote a short paper with Dimitar Jetchev in 2005 (which I forgot about until just now, and never published), which I think was motivated by this question: [[attachment::jetchev-stein-congruences_and_unramified_cohomology.pdf]]
 
People: ''Jared Weinstein'', Mirela Ciperiani, William Stein, Dimitar Jetchev, Ken Ribet, Barry Mazur

== Find an algorithm to decide if y_{p^n} is divisible by (g-1) and run it for a curve of rank >= 2 ==

   * Consider the Heegner $y_{p^n}$ over the anticyclotomic tower for a rank >= 2 curve, Sha trivial, etc. 

   * Barry Mazur remarks that this may be connected to his notion (with Rubin) of "Shadow lines" in Mordell-Weil. 

People: Mirela Ciperiani, William Stein, Barry Mazur, Jay Pottharst

== Compute Frobenius eigenvalues for a bunch of curves to illustrate Katz-Sarnak ==

 * curves that vary in various ways, e.g., coverings of one curve...

 * might use David Harvey's super fast Kedlaya zeta function implementation in Sage (his C++ code), but unfortunately the curves that typically arise aren't hyperelliptic, and Harvey's code only applies to hyperelliptic curves.

 * could lead to questions of an "anabelian sort"... e.g., take elliptic curve over finite field, take n-torsion points, extract roots, get coverings, etc.  Compute Frobenius eigenvalues of all these covers. 

 * To do a useful computation, one needs to compute Frobenius, say for the curve $y^{\ell} = f_N(x)$, where $f_N(x)$ is the $N$-th division polynomial on an elliptic curve.  This following paper is about how to count points on superelliptic curves: [[attachment:gaudry-gurel-an_extension_of_Kedlayas_point-counting_algorithm_to_superelliptic_curves.pdf|Gaudry-Gurel]]

 * There is a better paper maybe about this super-elliptic algorithm, with a Magma implementation here: http://www.math.tu-berlin.de/~minzlaff/

People: Barry Mazur, Kiran Kedlaya, Thomas Barnet-Lamb, David Harvey, Mirela Ciperiani, Sal Baig (lots of possibly relevant data over function fields)


== Sage Tutorials ==

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

   * Introduction to Python/Sage (Kiran Kedlaya, 10am on Wednesday)
   * Linear algebra modulo p (Robert Bradshaw, 11am 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 and William Stein, 11am on Friday)

Diff for "dayscambridge2/sprints"