Sage Days 18 Coding Sprint Projects

Contents

  1. Sage Days 18 Coding Sprint Projects
    1. Compute regulators of elliptic curves over function fields
    2. Is there an algorithm to enumerate all elliptic curves over a function field of a given conductor?
    3. Implement Tate's algorithm for elliptic curves over the function field $\mathbf{F}_p(t)$.
    4. Implement computation of the 3-Selmer rank of an elliptic curve over $\mathbf{Q}$.
    5. Compute statistics about distribution of Heegner divisors and Kolyvagin divisors modulo primes $p$.
    6. Create a table of images of Galois representations, for elliptic curves and/or Jacobians, in some range.
    7. Fully implement and optimize variant of Watkins's algorithm for fast computation of Heegner points.
    8. 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.
    9. Verify Kolyvagin's conjecture for a specific rank 3 curve.
    10. Implement an algorithm in Sage to compute Stark-Heegner points.
    11. Compute the higher Heegner point $y_5$ on the curve 389a '''provably correctly'''.
    12. Compute special values of the Gross-Zagier $L$-function $L(f,\chi,s)$.
    13. Implement something toward the Pollack et al. overconvergent modular symbols algorithm.
    14. Compute a Heegner point on the Jacobian of a genus 2 curve

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 $\mathbf{F}_p(t)$.

Implement computation of the 3-Selmer rank of an elliptic curve over $\mathbf{Q}$.

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

Create a table of images of Galois representations, for elliptic curves and/or Jacobians, in some range.

Fully implement and optimize variant of Watkins's algorithm for fast computation of Heegner points.

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.

Verify Kolyvagin's conjecture for a specific rank 3 curve.

Implement an algorithm in Sage to compute Stark-Heegner points.

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

Compute special values of the Gross-Zagier $L$-function $L(f,\chi,s)$.

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

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