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.
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 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 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;