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



