A bound on the $\mu $-invariants of supersingular elliptic curves

Abstract Let $E/\mathbb {Q}$ be an elliptic curve and let p be a prime of good supersingular reduction. Attached to E are pairs of Iwasawa invariants $\mu _p^\pm $ and $\lambda _p^\pm $ which encode arithmetic properties of E along the cyclotomic $\mathbb {Z}_p$ -extension of $\mathbb {Q}$ . A well-known conjecture of B. Perrin-Riou and R. Pollack asserts that $\mu _p^\pm =0$ . We provide support for this conjecture by proving that for any $\ell \geq 0$ , we have $\mu _p^\pm \leq 1$ for all but finitely many primes p with $\lambda _p^\pm =\ell $ . Assuming a recent conjecture of D. Kundu and A. Ray, our result implies that $\mu _p^\pm \leq 1$ holds on a density 1 set of good supersingular primes for E .