Tom Fisher

Let $A$ be any one of the three elliptic curves over ${\mathbb{Q}$ with conductor 11. We show that $A$ has Mordell--Weil rank zero over its field of 5-division points. In each case we also compute the 5-primary part of the Tate--Shafarevich group. Our calculations make use of the Galois equivariance of the Cassels--Tate pairing.

2000 Mathematical Subject Classification: 11G05, 11Y40, 11R23.

Keywords: elliptic curves, descent calculations, Cassels--Tate pairing, non-abelian Iwasawa theory.

E-mail: T.A.Fisher@dpmms.cam.ac.uk