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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? | == Compute regulators of elliptic curves over function fields == |
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| * Implement Tate's algorithm for elliptic curves over the function field $\mathbf{F}_p(t)$. | People: ''Sal Baig'' |
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| * Implement computation of the 3-Selmer rank of an elliptic curve over $\mathbf{Q}$. | == Is there an algorithm to enumerate all elliptic curves over a function field of a given conductor? == |
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| * Compute statistics about distribution of Heegner divisors and Kolyvagin divisors modulo primes $p$. | People: ''Sal Baig'', William Stein |
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| * Create a table of images of Galois representations, for elliptic curves and/or Jacobians, in some range. | |
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| * Fully implement and optimize variant of Watkins's algorithm for fast computation of Heegner points. | == Implement Tate's algorithm for elliptic curves over the function field F_p(t) == |
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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. | People: ''Sal Baig'', David Roe (?) |
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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. | == Implement computation of the 3-Selmer rank of an elliptic curve over QQ == |
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| * Compute the higher Heegner point $y_5$ on the curve 389a '''provably correctly'''. | People: ''Robert Miller'', William Stein |
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| * Compute special values of the Gross-Zagier $L$-function $L(f,\chi,s)$. | |
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| * Implement something toward the Pollack et al. overconvergent modular symbols algorithm. | == Compute statistics about distribution of Heegner divisors and Kolyvagin divisors modulo primes p == |
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| * Compute a Heegner point on the Jacobian of a genus 2 curve | People: ''William Stein'', Dimitar Jetchev == Create a table of images of Galois representations, for elliptic curves and/or Jacobians, in some range == People: ''Drew Sutherland'', William Stein Drew Sutherland remarks: {{{ Hi William, I would definitely be motivated to work on the table of Galois images project that you suggested in your list. I am currently rerunning my previous computations on the Stein-Watkins database using an improved version of the algorithm (just for the mod ell case at the moment, I still want to tweak the mod ell^k code some more). It would be great to get all this data organized and accessible in a useful form, especially while everything is fresh in my mind. Drew }}} == Fully implement and optimize variant of Watkins's algorithm for fast computation of Heegner points == People: ''William Stein'', Robert Bradshaw == 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 == People: ''Jared Weinstein'' == Verify Kolyvagin's conjecture for a specific rank 3 curve == People: ''William Stein'' == Implement an algorithm in Sage to compute Stark-Heegner points == People: ''Mathew Greenberg'' == Compute the higher Heegner point y_5 on the curve 389a provably correctly == People: ''Robert Bradshaw'', William Stein == Compute special values of the Gross-Zagier L-function L(f,chi,s) == People: ''Robert Bradshaw'' == Implement something toward the Pollack et al. overconvergent modular symbols algorithm == People: ''Robert Pollack'', Avner Ash == Compute a Heegner point on the Jacobian of a genus 2 curve == People: ? == Visibility of Kolyvagin Classes == People: ''Jared Weinstein'', William Stein This might be idle blather, but I've been thinking about Kolyvagin classes and I'm curious about the following. 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. |
Sage Days 18 Coding Sprint Projects
Contents
-
Sage Days 18 Coding Sprint Projects
- 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
- Visibility of Kolyvagin Classes
Compute regulators of elliptic curves over function fields
People: Sal Baig
Is there an algorithm to enumerate all elliptic curves over a function field of a given conductor?
People: Sal Baig, William Stein
Implement Tate's algorithm for elliptic curves over the function field F_p(t)
People: Sal Baig, David Roe (?)
Implement computation of the 3-Selmer rank of an elliptic curve over QQ
People: Robert Miller, William Stein
Compute statistics about distribution of Heegner divisors and Kolyvagin divisors modulo primes p
People: William Stein, Dimitar Jetchev
Create a table of images of Galois representations, for elliptic curves and/or Jacobians, in some range
People: Drew Sutherland, William Stein
Drew Sutherland remarks:
Hi William, I would definitely be motivated to work on the table of Galois images project that you suggested in your list. I am currently rerunning my previous computations on the Stein-Watkins database using an improved version of the algorithm (just for the mod ell case at the moment, I still want to tweak the mod ell^k code some more). It would be great to get all this data organized and accessible in a useful form, especially while everything is fresh in my mind. Drew
Fully implement and optimize variant of Watkins's algorithm for fast computation of Heegner points
People: William Stein, Robert Bradshaw
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
People: Jared Weinstein
Verify Kolyvagin's conjecture for a specific rank 3 curve
People: William Stein
Implement an algorithm in Sage to compute Stark-Heegner points
People: Mathew Greenberg
Compute the higher Heegner point y_5 on the curve 389a provably correctly
People: Robert Bradshaw, William Stein
Compute special values of the Gross-Zagier L-function L(f,chi,s)
People: Robert Bradshaw
Implement something toward the Pollack et al. overconvergent modular symbols algorithm
People: Robert Pollack, Avner Ash
Compute a Heegner point on the Jacobian of a genus 2 curve
- People: ?
Visibility of Kolyvagin Classes
People: Jared Weinstein, William Stein
This might be idle blather, but I've been thinking about Kolyvagin classes and I'm curious about the following. 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.