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

TL;DR

This paper establishes a precise criterion linking the Iwasawa $mbda$-invariant of an imaginary quadratic field's cyclotomic extension to the divisibility of points on a related elliptic curve, enhancing understanding of number theory invariants.

Contribution

It provides a new characterization of the Iwasawa $mbda$-invariant in terms of elliptic curve point divisibility for imaginary quadratic fields.

Findings

Iwasawa's $mbda$-invariant exceeds 1 iff the elliptic curve's points are divisible by $p^2$

The criterion applies to primes $p > 3$ not dividing the class number of $K$

Connects Iwasawa theory with elliptic curve point divisibility in quadratic fields.

Abstract

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 $p^2$.