Sage Days 18 Coding Sprint Projects

Elliptic curves over function fields

This project will include the following topics:

y^2 = x^3 + \left(3 t^{6} + 3 t^{5} + 3 t^{3} + 3 t^{2}\right)x +\left(4 t^{9} + t^{8} + 3 t^{7} + 2 t^{6} + 3 t^{5} + t^{4} + 4 t^{3}\right).

The L-funtion of E_f is:

L(E_f/K,T) = 1-5T-25T^2+125T^3.

The analytic rank is 2 with L^{(2)}(E_f/K,1/5) = 4. The points P=[3t^3 + t^2 + 4t: t^3 + 3t^2:1], Q=[3t^3 + 2t^2 + 3t: t^3:1] are independent on E_F(K), so its algebraic rank is 2. Thus the refined BSD conjecture is true. We do not if P,Q generate E_f(K) mod torsion, so the determinant of their height matrix (=5) gives an upper bound on the regulator of E_f/K. The Tamagawa number was computed by hand to be 2^7/5. The torsion subgroup has size 4 (all 2-torsion). Thus

\frac{4}{2} = \frac{L^{(r)}(E_f/K,1/q)}{r!} = \frac{|Sha|R\tau}{|E(K)_{tors}|^2}\leq\frac{|Sha|\cdot 5\cdot 2^7\cdot 5^{-1}}{2^4}.

Note that there is probably a factor of 2 going unaccounted for in the computation of the regulator (as well as a \log q factor possibly). From this, we see that |Sha|\geq \frac{1}{4} (again, this is probably off by a power of 2, suggesting the bound should really be 1).

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:

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

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:

People: Drew Sutherland, Ken Ribet, William Stein, Kiran Kedlaya, David Roe

Fast computation of Heegner points

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

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.

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:

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

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

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

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

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

Frobenius eigenvalues, and my worry is that one won't really see anything terribly interesting if one works only with N=2; but maybe when one works with N=3. For example, take an elliptic curve E over F_q and let f_3 be (3-division") function on E: meaning that it vanishes to order one at all nontrivial 3-torsion points and has the appropriate order pole at the origin (and no other poles or zeroes). Then (for small n) extract n-th powers of that f_3 to get curves C_n mapping to E (of unfortunately high genus). But it could be that the collection of their Frobenius eigenvalues tells us more than just the isogeny class of E? From what I learn by talking with William, this might be a very difficult computational problem though..."

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:

dayscambridge2/sprints (last edited 2009-12-23 06:38:37 by was)