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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. 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 $\mathbb{F}_q(t)$.
      [[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WKD-4HVF15H-2&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=ec0b143bb1c45f3bdcd4bea8841e8735|Diophantine Equations over Global Function Fields I: The Thue Equation]]
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 * Create a table of images of Galois representations, for elliptic curves and/or Jacobians, in some range.       [[http://akpeters.metapress.com/content/458402lu66634164/|Diophantine Equations over Global Function Fields II: R-Integral Solutions of Thue Equations]]
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 * Fully implement and optimize variant of Watkins's algorithm for fast computation of Heegner points.       [[http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.facm/1229696556&page=record|Diophantine equations over global function fields III: An application to resultant form equations]]
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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.       [[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WKD-4TP7HDM-3&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=c50fd011b673de5511d49b10b3fe9935|Diophantine equations over global function fields IV: S-unit equations in several variables with an application to norm form equations]]
      The paper [[http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.em/1204928531|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).
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 * Verify Kolyvagin's conjecture for a specific rank 3 curve.  * Implement Tate's algorithm for elliptic curves over the function field $\mathbb{F}_p(t)$.
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 * Implement an algorithm in Sage to compute Stark-Heegner points.  * Verify BSD for elliptic curves over function fields of analytic rank 3 or higher
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 * Compute the higher Heegner point $y_5$ on the curve 389a '''provably correctly'''.  * Robert Bradshaw implemented faster arithmetic for $F_p(t)$: [[http://trac.sagemath.org/sage_trac/ticket/7585|#7585]]
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 * Compute special values of the Gross-Zagier $L$-function $L(f,\chi,s)$.  * 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
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 * Implement something toward the Pollack et al. overconvergent modular symbols algorithm. $$ y^2 = x^3 + \left(3 t^{6} + 3 t^{5} + 3 t^{3} + 3 t^{2}\right)x +\left(4 t^{9} + t^{8} + 3 t^{7} + 2 t^{6} + 3 t^{5} + t^{4} + 4 t^{3}\right). $$
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 * Compute a Heegner point on the Jacobian of a genus 2 curve The L-funtion of $E_f$ is:

$$ L(E_f/K,T) = 1-5T-25T^2+125T^3.$$

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

$$ \frac{100}{2} = \frac{L^{(r)}(E_f/K,1/q)}{r!} = \frac{|Sha|R\tau}{|E(K)_{tors}|^2}\leq\frac{|Sha|\cdot 5\cdot 2^7\cdot 5^{-1}}{2^4}.$$

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).

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: [[attachment:ThreeDescent.gp|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 [[attachment:three_descent.sage|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: [[/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!

  * 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 [[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]] 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)

Sage Days 18 Coding Sprint Projects

Elliptic curves over function fields

This project will include the following topics:

y^2 = x^3 + \left(3 t^{6} + 3 t^{5} + 3 t^{3} + 3 t^{2}\right)x +\left(4 t^{9} + t^{8} + 3 t^{7} + 2 t^{6} + 3 t^{5} + t^{4} + 4 t^{3}\right).

The L-funtion of E_f is:

L(E_f/K,T) = 1-5T-25T^2+125T^3.

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

\frac{100}{2} = \frac{L^{(r)}(E_f/K,1/q)}{r!} = \frac{|Sha|R\tau}{|E(K)_{tors}|^2}\leq\frac{|Sha|\cdot 5\cdot 2^7\cdot 5^{-1}}{2^4}.

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).

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 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)

dayscambridge2/sprints (last edited 2009-12-23 06:38:37 by was)