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

TL;DR

This paper investigates the Iwasawa invariants of supersingular elliptic curves, providing bounds on the $d$-invariants and supporting conjectures about their vanishing for almost all primes.

Contribution

It establishes an upper bound of 1 for the $d$-invariants for all but finitely many primes, supporting the conjecture that these invariants vanish.

Findings

d$-invariants are bounded above by 1 for almost all primes

Supports the conjecture that d$-invariants are zero for most primes

Results hold under certain conjectural assumptions

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 $E$.