Heegner-like phenomena for elliptic curves in non-commutative p-adic Lie 
                              extensions

                               John Coates
                        University of Cambridge


I will discuss a strange blend of Iwasawa theory and behaviour of the
complex L-functions of elliptic curves at s=1 up certain non-commutative
towers which Sujatha and I have recently discovered. In these towers, we
prove that the multiplicity of the zero at s=1 of the complex L-function
(the analyic rank) tends uniformly to infinity as one mounts the tower,
and that the rank of the Mordell-Weil group is bounded above always by the
analytic rank; moreover if the analytic rank is equal to the rank always,
we show that the p-primary subgroup of the Tate group is always finite in
the tower. I will also discuss some numerical examples which follow from
the calculations of Tim and Vladimir Dokchitser.