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, 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.

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: 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

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, 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.

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)
• 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.

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)