Abstract view
On the Vanishing of $\mu$Invariants of Elliptic Curves over $\qq$


Published:20050801
Printed: Aug 2005
Abstract
Let $E_{/\qq}$ be an elliptic curve with good ordinary reduction at a
prime $p>2$. It has a welldefined Iwasawa $\mu$invariant $\mu(E)_p$
which encodes part of the information about the growth of the Selmer
group $\sel E{K_n}$ as $K_n$ ranges over the subfields of the
cyclotomic $\zzp$extension $K_\infty/\qq$. Ralph Greenberg has
conjectured that any such $E$ is isogenous to a curve $E'$ with
$\mu(E')_p=0$. In this paper we prove Greenberg's conjecture for
infinitely many curves $E$ with a rational $p$torsion point, $p=3$ or
$5$, no two of our examples having isomorphic $p$torsion. The core
of our strategy is a partial explicit evaluation of the global duality
pairing for finite flat group schemes over rings of integers.