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 F_p(t)
    4. Implement computation of the 3-Selmer rank of an elliptic curve over QQ
    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 F_p(t)

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

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