= L-series =

<<TableOfContents>>

=== Project Leader ===

William

=== Group Members ===

William, Cassie, Amy, Lola

=== Getting Going on Development on *your* laptop (not *.sagenb.org) ===

 1. Download this file: http://sage.math.washington.edu/home/wstein/days/33/eulerprod.py

 2. Type in the notebook: {{{attach /exact/path/to/eulerprod.py}}}

 3. Note that elliptic curves over Q(sqrt(5)) won't work if you don't have psage installed. 

=== Project Description ===

For L-series lovers:  Getting the doctest coverage to 100% on this
might be a good project:

  http://code.google.com/p/purplesage/source/browse/psage/lseries/eulerprod.py

See [[http://nt.sagenb.org/home/pub/144/|this worksheet for an example]].


That may discover "issues" (bugs), which I would likely have to fix,
but would also be fun because one gets to come up with lots of
creative examples of L-series all over the place.   Also, the top of
that file has a todo list for new features to implement -- most would
be bad projects, but one which would be good would be to make it so
the Lseries object can use Lcalc (Rubinstein's program) to compute
L-series instead of Dokchitser.  This would be a good project, because
it would mainly involve thinking about the annoying mathematics
involved in going between normalizing L-series with the center of the
critical strip at 1/2 versus not doing that.  Also, it is all pure
Python, so easier to get going.

Anyway, I'd say 1 could be a good project for people who know the
basics of L-series, but want to get a much more concrete feel for
them.  In fact, instead of just trying to get coverage to 100%,
writing a *tutorial* for computing with L-series using that package
would be really nice.   E.g., one could walk through how to find
missing information, create new L-series classes, etc.

===  Specific Concrete Projects ===


     * Move eulerprod.py into Sage itself.  This would at least involve:
          1. Clean up the imports at the top of the file.
          2. Do something with: "from psage.ellcurve.lseries.helper import extend_multiplicatively_generic"; it is the main dependency.  It's only a page of code. 
          3. Graceful failure when 
     * Doctest coverage to 100%
     * Fix bugs that are found when doctesting.
     * Use rubinstein _lcalc (see line 1116)
     * Math question: given that you have a_p for Norm(p) <= B, how many bits of precision do we get?  Basically, invert the function L.number_of_coefficients(prec=50)
     * Write an L-functions tutorial:  http://sagemath.org/doc/thematic_tutorials   This does not have to be long.  It has to start somewhere.  Small and focused first is fine!
     * Speed of elliptic curves over number fields (smalljac).
     * Make it so we use exactly one GP session for *all* of the Dokchitser L-functions
     * Symmetric powers (and modular degree -- see trac 9758)
     * Triple product L-functions: Gross-Kudla, Zhang, etc -- see the code in triple_prod/triple.py
     * Make it so we use exactly one GP session for *all* of the Dokchitser L-functions
     * Tensor products
     * Genus 2 curves, via smalljac and genus2reduction
     * Fast L-series of elliptic curves over number fields (not just sqrt(5)), via smalljac
     * Inverse of number_of_coefficients function.     

=== Work in progress ===

L-series tutorial: [[http://wiki.sagemath.org/days33/lfunction/tutorial]]

Triple product L-function code: [[http://wiki.sagemath.org/days33/lfunction/tripleproduct]]