A bound on the $μ$-invariants of supersingular elliptic curves

Rylan Gajek-Leonard

Published 2024-09-26, updated 2024-12-19Version 2

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 $\lambda_p^\pm=\ell$. As a corollary, we show that if $E$ has Mordell-Weil rank 0 then $\mu_p^\pm\leq 1$ holds on density 1 set of good supersingular primes.