On CM elliptic curves and the cyclotomic $\lambda$-invariants of imaginary quadratic fields

Let $K$ be an imaginary quadratic field, and fix a prime $p>3$ that does not divide the class number of $K$. In this paper we prove that Iwasawa’s $\lambda$-invariant for the cyclotomic ${\mathbb{Z}}_p$-extension of $K$ is greater than $1$ if and only if the number of points on a certain reduced elliptic curve is divisible by $p^2$.