todo

TODO LIST

[x] (andrew) table of how many isogeny classes have each prime isogeny degree

[x] (andrew) clean up the table's section a bit (probably just want
    to write a few sentences at the beginning that say something like
    "we list a bunch of tables with..."

[x] (william) descriptions of tables

[x] (william) make sure each section has a topic paragraph

[x] (Ben) Need a better reference to give all families than Kubert?

[x] things to have tables of (or information about):
    - # of curves with each torsion group (mostly done for 199); but
      redo for each torsion structure (not order), e.g., write as 
          1, 2, 3, 4, 2x2, 5, 6, 7, 8, 2x4, 9, etc. 
    - # of curves with each rank (may not need to be a separate table,
        since it is not much information (done for 199)
    - # of isogeny classes of each size (done for 199)
    (everybody) Other interesting statistics?
    - maybe a table with the number of isogeny/isomorphism
      classes of curves with norm conductor <= N * 100, for N = 1 .. 18,
      or something like that
    - information about the distribution of a_p for a few small p, i.e.,
      for the curves in our data set, here is how many have a_2 = -2,
      -1, 0, 1, 2, etc.

[x] (aly, andrew) include more tables, with information up to rank 2

[x] (andrew) How was curve found in "Extremely naive enumeration" section?

[x] (andrew) Is it really impossible to find the example using
L-functions (very end of section 3) using mod-jon?

[X] (andrew) Is it really impossible to find the example using
L-functions (very end of section 3) using mod-jon? 
A. Yes it really is impossible

[X] (everybody) Figure out problems with isogeny code, then draw an
    isogeny graph

[X] (william) fill in acknowledgement for NSF

[X] (anybody) do something about table 6.5 (CM curves), because it is
    too wide; e.g., put the j-invariants in a different table or the
    text, since there are only 2 of them, and they are repeated a
    bunch.  Then use extra space to give the *actual* conductor (not
    just the norm).

[X] (mainly ashwath) table of all curves with nontrivial conjectural sha.
    as corollary -- BSD appears to be true for these curves. we mostly
    checked the full conjecture, right, using the L-function to compute
    the order of sha? (By "we", I mean "someone", but not me.)

[x] (anybody) example where Cremona-Lingham is useful.

[x] (william) write section 2.2.3

[x] (william) write section 2.2.4

[x] (william) write section 2.2.5.  Note that there is a theoretical
    obstruction to doing this provably correctly at present!

[x] (william or ben or ashwath) fill in proposition 3.1: " TODO: Explain how to tell if $\ell\mid \#E(F)_{\tor}$."

[x] Put the example in Section 3.1 of the biggest "height" curve (but
smallest in isogeny class) up to norm conductor 1831.  -- Concrete
question: To illustrate the difficulty of using naive enumeration to
find a curve E_f associated to f, it would be nice to give the worst
example.  So if we take the best curve from each isogeny class up to
1831, what is the worst one, i.e., with "biggest coefficients" in some
sense?

[x] (william) add a remark to residue rings section explaining why our
    choice of representation is adapted to this problem.

[x] (william) fix notation in residue ring section to not conflict. 

[x] (jon) reference for Dembele's "An algorithm for modular elliptic curves..." in 3.6

[x] (william) delete section 6.3

[x] (william) bottom of page 1, "is supported by theoretical evidence"
      (citation needed?) -- can copy something from my NSF proposal

[x] clean up and update references; grabbing bibtex entries from mathscinet

[x] (jon) reference and comments for conjecture 3.2 need to be fixed

[x] (william or Jon) write an abstract