Computing the Iwasawa $\mu$-Invariants for Elliptic Curves over ${\mathbb{Z}}_2$-Extensions

Abstract

Let $E$ be an elliptic curve defined over rational numbers. Further assume $E$ has good ordinary reduction at $p=2$ and $E[4]$ is reducible as a $G_{\mathbb{Q}}$-representation. In this paper, we offer sufficient and necessary computational criteria for the algebraic Iwasawa $\mu$-invariant over the cyclotomic ${\mathbb{Z}}_2$-extension of rational number to be $0$, $1$ or $\ge 2$. We also show that elliptic curves with isomorphic mod-$4$ representations have algebraic $\mu$-invariant equal to $0$ if one of the curves does.

arXiv comment27 pages. Comments welcome!