A bound on the $μ$-invariants of supersingular elliptic curves
Rylan Gajek-Leonard
TL;DR
The paper addresses the μ-invariant bounds for supersingular elliptic curves along the cyclotomic $ ext{Z}_p$-extension by connecting algebraic and analytic Iwasawa invariants through the main conjecture and by leveraging Mazur–Tate elements and modular symbols. The authors establish that for any $oldsymbol{ ext{ℓ}oldsymbol{ix}}$, the signed μ-invariants satisfy $oldsymbol{ ext{μ}_{p}^{oldsymbol{ullet}} oldsymbol{ rianglelefteq} 1}$ for all but finitely many good supersingular primes with $oldsymbol{ ext{λ}_{p}^{oldsymbol{ullet}}=oldsymbol{ℓ}}$, and show how this bound can be extended to a density-1 set under the Kundu–Ray conjecture. The approach hinges on refined Iwasawa invariants, two equivalent definitions (division-algorithm and augmentation-ideal), and the analysis of Mazur–Tate elements $oldsymbol{ heta_n}$ linked to the signed $p$-adic $L$-functions $oldsymbol{L_p^{oldsymbol{ullet}}}$. The results provide evidence for the vanishing of μ-invariants in the supersingular setting and support the Perrin-Riou–Pollack conjecture on the μ-invariant, with implications for the structure of signed Selmer groups and main conjectures in Iwasawa theory.
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 $μ_p^\pm$ and $λ_p^\pm$ 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 $μ_p^\pm=0$. We provide support for this conjecture by proving that for any $\ell\geq 0$, we have $μ_p^\pm\leq 1$ for all but finitely many primes $p$ with $λ_p^\pm=\ell$. Assuming a recent conjecture of D. Kundu and A. Ray, our result implies that $μ_p^\pm\leq 1$ holds on a density 1 set of good supersingular primes for $E$.