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# On the Vanishing of $\mu$-Invariants of Elliptic Curves over $\qq$

Let $E_{/\qq}$ be an elliptic curve with good ordinary reduction at a prime $p>2$. It has a well-defined 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.