On Conjectures of $\mu$-Invariant for Selmer Groups at $p=2$

Abstract

Let $E$ be an elliptic curve define over $\mathbb{Q}$. Further assume $E$ has good ordinary reduction at $p=2$. In this article, we prove Greenberg's conjecture on the value of the algebraic Iwasawa $\mu$-invariant when $E$ has a rational isogeny of degree 2, and obtain a sharp absolute upper bound of the $2$-adic $\mu$-invariants for all such curves. We also prove the vanishing of the algebraic $\mu$-invariant of the fine Selmer group introduced by Coates and Sujatha at $p=2$.