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| * Compute regulators of elliptic curves over function fields | <<TableOfContents>> |
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| * Is there an algorithm to enumerate all elliptic curves over a function field of a given conductor? | == Elliptic curves over function fields == |
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| * Implement Tate's algorithm for elliptic curves over the function field $\mathbf{F}_p(t)$. | This project will include the following topics: |
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| * Implement computation of the 3-Selmer rank of an elliptic curve over $\mathbf{Q}$. | * 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/ |
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| * Compute statistics about distribution of Heegner divisors and Kolyvagin divisors modulo primes $p$. | * 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. |
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| * Create a table of images of Galois representations, for elliptic curves and/or Jacobians, in some range. | * Implement Tate's algorithm for elliptic curves over the function field F_p(t) |
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| * Fully implement and optimize variant of Watkins's algorithm for fast computation of Heegner points. | * Verify BSD for elliptic curves over function fields of analytic rank 3 or higher |
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| * 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. | * Robert Bradshaw implemented faster arithmetic for $F_p(t)$: [[http://trac.sagemath.org/sage_trac/ticket/7585|#7585]] |
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| * Verify Kolyvagin's conjecture for a specific rank 3 curve. | |
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| * Implement an algorithm in Sage to compute Stark-Heegner points. | People: ''Sal Baig'', William Stein, David Roe, Ken Ribet, Kiran Kedlaya, Robert Bradshaw, Victor Miller (S-unit equations), Thomas Barnet-Lamb |
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| * Compute the higher Heegner point $y_5$ on the curve 389a '''provably correctly'''. | |
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| * Compute special values of the Gross-Zagier $L$-function $L(f,\chi,s)$. | == Implement computation of the 3-Selmer rank of an elliptic curve over QQ == |
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| * Implement something toward the Pollack et al. overconvergent modular symbols algorithm. | Some projects: * Implement the algorithm from Schaefer-Stoll which does the general case |
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| * Compute a Heegner point on the Jacobian of a genus 2 curve | * Implement the algorithm in the reducible case due to Jeechul Woo: [[attachment:ThreeDescent.gp|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. 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: [[/5077aestimates|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! 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 [[http://groups.google.com/group/sageday18/browse_thread/thread/6904bf31c59bb444|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: [[attachment::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: [[attachment:gaudry-gurel-an_extension_of_Kedlayas_point-counting_algorithm_to_superelliptic_curves.pdf|Gaudry-Gurel]] * There is a better paper maybe about this super-elliptic algorithm, with a Magma implementation here: http://www.math.tu-berlin.de/~minzlaff/ 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) |
Sage Days 18 Coding Sprint Projects
Contents
-
Sage Days 18 Coding Sprint Projects
- Elliptic curves over function fields
- 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
- Fast computation of Heegner points
- Implement code to compute the asymptotic distribution of Kolyvagin classes
- 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 a Heegner point on the Jacobian of a genus 2 curve
- Visibility of Kolyvagin Classes
- Find an algorithm to decide if y_{p^n} is divisible by (g-1) and run it for a curve of rank >= 2
- Compute Frobenius eigenvalues for a bunch of curves to illustrate Katz-Sarnak
- Sage Tutorials
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.
- 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
Robert Bradshaw implemented faster arithmetic for F_p(t): #7585
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. 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!
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
There is a better paper maybe about this super-elliptic algorithm, with a Magma implementation here: http://www.math.tu-berlin.de/~minzlaff/
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)