## On the Iwasawa $\mu $-invariants of supersingular elliptic curves

### Volume 194 / 2020

Acta Arithmetica 194 (2020), 179-186
MSC: Primary 11R23; Secondary 11G05.
DOI: 10.4064/aa190213-28-8
Published online: 2 March 2020

#### Abstract

We explore the relation between the Iwasawa invariants $\mu ^{+}$ and $\mu ^{-}$ associated respectively with the plus and the minus Selmer groups of two elliptic curves $E_{1}$ and $E_{2}$ over $\mathbb {Q}$ having isomorphic Galois representations $E_{1}[p^{r}]\cong E_{2}[p^{r}]$ at a prime $p$ of supersingular reduction. We prove that $\mu ^{\pm }(E_{1})=\mu ^{\pm }(E_{2})$ if either is less than $r$, and $\mu ^{\pm }(E_{1}), \mu ^{\pm }(E_{2})\geq r $ if either is greater than or equal to $r$.