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. It's well known that finding all Elliptic Curves with good reduction outside a finite set, S of places is equivalent to solving S-unit equations (this is not true when the curve is iso-trivial, though that's not much of a problem here). There are a series of four papers by Gaal and Pohst that give efficient algorithms for this when dealing with function fields over finite fields, especially \mathbb{F}_q(t).

    • Diophantine Equations over Global Function Fields I: The Thue Equation

      Diophantine Equations over Global Function Fields II: R-Integral Solutions of Thue Equations

      Diophantine equations over global function fields III: An application to resultant form equations

      Diophantine equations over global function fields IV: S-unit equations in several variables with an application to norm form equations The paper Finding All Elliptic Curves with Good Reduction Outside a Given Set of Primes by Cremona and Lingham has a lot of details and references to the analogous problem over number fields (which should be harder).

• Implement Tate's algorithm for elliptic curves over the function field \mathbb{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

• Robert Bradshaw has implemented fast point search: http://sagenb.org/home/pub/1228/

• Let K=\mathbb{F}_5(t) and E/K: y^2 = x(x+t)(x+t^2) be the Legendre curve and consider its twist by f=1+t, E_f. E_f has Weierstrass equation

The L-funtion of E_f is:

The analytic rank is 2 with L^{(2)}(E_f/K,1/5) = 100. 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

Note that there is probably a factor of 2 going unaccounted for in the computation of the regulator. From this, we see that |Sha|\geq \frac{25}{4} (again, this may be off by a power of 2).

• Other curves of analytic rank 2:

  • The Legendre curve y^2 = x(x+t)(x+t^2)/\mathbb{F}_{11}(t) twisted by t+7 with L-function:

    • 1 -11T -121T^2+ 1331T^3

  • The Legendre curve y^2 = x(x+t)(x+t^2)/\mathbb{F}_{5}(t) twisted by t^3+1 with L-function:

    • 1+ 3T -15T^2 -125T^3 -625T^4 -1875T^5 +9375T^6 +78125T^7
• Some analytic rank 3 curves:

  • The Legendre curve y^2 = x(x+t)(x+t^2)/\mathbb{F}_{5}(t) twisted by t^5+t^2+t+1 with L-function:

    • 1 +T -25T^2 -121T^3 -190T^4 -150T^5 +750T^6 +23750T^7+ 378125T^8+ 1953125T^9 -1953125T^10 -48828125T^11

  • The Legendre curve y^2 = x(x+t)(x+t^2)/\mathbb{F}_{7}(t) twisted by t^3+t^2+t+1 with L-function:

    • 1 -9T+ 29T^2 -805T^3+ 5635T^4 -9947T^5+ 151263T^6 -823543T^7

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

  • Progress: Computation of S-class groups and S-units is now available for etale algebras: S_units_for_etale_algebras.patch

• Implement the algorithm in the reducible case due to Jeechul Woo: Jeechul Woo's GP script Here is a sage worksheet with the code attached and usable with Sage: http://sagenb.org/home/pub/1200/ and a sage port.

• 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.
• Status report on Thursday from Drew: "I now have a standalone version mostly working that is driven entirely by precomputed tables. It's about 6-7 times faster and can crank through the entire Stein-Watkins database in under 2 hours. Still needs a bit of debugging, which I will continue working on tonight."

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!

• Using the algorithm from Jared's talk we computed and found that the first tau we can easily try should work for verifying Kolyvagin's conjecture. Code here: http://sagenb.org/home/pub/1203

• Jen and William (Thursday night):

  • computed \tau_n for the rank 3 curve and many n using the algorithm from Jared Weinstein's talk

  • implemented code to compute the Kolyvagin sigma operator on c that are a squarefree product of primes (and not just on primes). the first 2 versions/algorithms were *wrong*, but we found something in the end and implemented it We ran it and verified that a Kolyvagin class is nonzero for the rank 3 curve 5077a. We also tried this with some different n and got consistent results. So Koly's conjecture appars to hold 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:

• 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 More from Barry: " I 've thought about my off-the-cuff suggestion for computing

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..."

• There is a better paper maybe about this super-elliptic algorithm, with a Magma implementation here: http://www.math.tu-berlin.de/~minzlaff/. And here is a Sage worksheet that *wraps* that Magma code, so if you have Magma you can compute quickly the zeta function of y^\ell = f(x) for f of odd degree. And indeed Minzlaff's code seems to work fine.

• Kiran has a short Sage program to compute the zeta function of y^4 = (cubic), to be published soon.

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)