Tuesday 29 May 2007, MR13, 2:30
Finding rational points on elliptic curves using 6-descent and 12-descent
Descent on an elliptic curve E is used to obtain partial information about
both the group of rational points (the Mordell-Weil group) and the failure
of the Hasse principle for certain twists of E (the Tate-Shafarevich group).
The Selmer group elements computed may be represented as n-coverings of E.
Traditionally one takes n to be a prime power. Breaking with this tradition,
I explain how to combine the data of an m-covering and an n-covering,
for m and n coprime, to obtain an mn-covering. This technique improves
the search for rational points on E. In particular using 6-descent and 12-descent,
I was recently able to find all the "missing" generators for the elliptic
curves of analytic rank 2 in the Stein-Watkins database.