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| <<TableOfContents>> == Elliptic curves over function fields == This project will include the following topics: |
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| * Implement Tate's algorithm for elliptic curves over the function field $\mathbf{F}_p(t)$. | * Implement Tate's algorithm for elliptic curves over the function field F_p(t) |
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| * Implement computation of the 3-Selmer rank of an elliptic curve over $\mathbf{Q}$. | * Verify BSD for elliptic curves over function fields of analytic rank 3 or higher |
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| * Compute statistics about distribution of Heegner divisors and Kolyvagin divisors modulo primes $p$. | |
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| * Create a table of images of Galois representations, for elliptic curves and/or Jacobians, in some range. | People: ''Sal Baig'', William Stein, David Roe, Ken Ribet, Kiran Kedlaya, Robert Bradshaw, Victor Miller (S-unit equations) |
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| * Fully implement and optimize variant of Watkins's algorithm for fast computation of Heegner points. | |
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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. | == Implement computation of the 3-Selmer rank of an elliptic curve over QQ == |
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| * Verify Kolyvagin's conjecture for a specific rank 3 curve. | Some projects: * Implement the algorithm from Schaefer-Stoll which does the general case |
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| * Implement an algorithm in Sage to compute Stark-Heegner points. | * Implement the algorithm in the reducible case due to Jeechul Woo. |
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| * Compute the higher Heegner point $y_5$ on the curve 389a '''provably correctly'''. | * Compute the 3-Selmer ranks of all curves of conductor up to 1000 |
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| * Compute special values of the Gross-Zagier $L$-function $L(f,\chi,s)$. | People: ''Robert Miller'', William Stein, Victor Miller, Jeechul Woo (Noam's student; around only Thu, Fri) |
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| * Implement something toward the Pollack et al. overconvergent modular symbols algorithm. | |
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| * Compute a Heegner point on the Jacobian of a genus 2 curve | == 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. 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. 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. * 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 == 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 == 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 == 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. 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. 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) * 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. 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
- 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)
- Verify BSD for elliptic curves over function fields of analytic rank 3 or higher
People: Sal Baig, William Stein, David Roe, Ken Ribet, Kiran Kedlaya, Robert Bradshaw, Victor Miller (S-unit equations)
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.
- 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.
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.
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.
- 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
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
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
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.
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.
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)
- 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.
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)