Greenberg's μ=0 conjecture for elliptic curves over ℚ

Let $E/\mathbf{Q}$ be an elliptic curve, and let $p$ be an odd prime of good ordinary reduction. The representation $E[p]$ is irreducible as a Galois module when it has no nontrivial Galois-stable subspaces. Greenberg's μ=0 conjecture. If $E[p]$ is irreducible, then the $\mu$-invariant of $E$ over the cyclotomic $\mathbf{Z}_p$-extension of $\mathbf{Q}$ satisfies $\mu=0$. This generalizes the classical $\mu=0$ conjecture to elliptic curves; it is a longstanding open problem, although the paper proves $\mu\leq 1$ for all but finitely many good ordinary primes.