= Elliptic Curves: Specific Task List =

Let a be a root of $x^2-x-1$.

== Specific Concrete Little Questions ==

  * Find the isogeny class of the curve [1,a+1,a,a,0].
  * Data up to norm conductor 124.

== Major Projects ==
I keep thinking of good ideas for projects for the summer REU involving elliptic curves.  Thus I'll start listing them here.  -- William

  * Create a SQLite version of [[http://www.warwick.ac.uk/~masgaj/ftp/data/|Cremona's tables of elliptic curves]].  Make indexes and have it be very easy to query.  Have it replace the current Cremona database in Sage.

  * Work out the details of the analogue of the *statement* of the Gross-Zagier formula for elliptic curves over Q(sqrt(5)).  Use this to give an algorithm to compute the height of Heegner points, hence their index $[E(K): Z y_K]$ in the full group of rational points.

  * Give the analogue of Kolyvagin's bound -- that sqrt(#Sha) divides $[E(K): Z y_K]$ over Q(sqrt(5)).  Use this to prove that some Shafarevich-Tate groups of elliptic curves over Q(sqrt(5)) are trivial.  This will involve Galois cohomology and following -- perhaps word for word (!) -- the arguments in either B. Gross or W. McCallum's articles on Kolyvagin's work.  

  * Create as complete as possible of tables like Cremona's electronic tables, but for elliptic curves over Q(sqrt(5)).  

  * Create a *print* version of tables of all curves up to norm conductor 1000 over Q(sqrt(5)), which looks just like Cremona's tables. 

  * Give an algorithm to find minimal twists over Q(sqrt(5)) that is exactly analogous to the one described in Section 2.1 of [[http://wstein.org/papers/stein-watkins/ants.pdf|Stein-Watkins]].

  * Give an analogue of Mazur's theorem over Q(sqrt(5)). This is the theorem that classifies the prime degrees of rational isogenies. (Update: A different but related problem has been solved by Filip Najman -- personal email to wstein on June 1, 2011.)  http://web.math.hr/~fnajman/skfn.pdf;