On Gauss factorials and their connection to the cyclotomic $\lambda$-invariants of imaginary quadratic fields

Abstract

In this paper we establish a connection between the Gauss factorials and Iwasawa's cyclotomic $\lambda$-invariant for an imaginary quadratic field $K$. As a result, we will explain a corespondance between the 1-exceptional primes of Cosgrave and Dilcher for $m = 3$ and $m = 4$, and the primes for which the $\lambda$-invariants for $K = \mathbb{Q}(\sqrt{-3})$ and $K = \mathbb{Q}(i)$ is greater than one, respectively. We refer to the latter primes as ``non-trivial'' for their respective fields. We will also see that similar correspondences are true for $K = \mathbb{Q}(\sqrt{-d})$ when $d = 2,5$ and $6$. As a corollary we find that primes $p$ of the form $p^2 = 3x^2 + 3x + 1$ are always non-trivial for $K = \mathbb{Q}(\sqrt{-3})$. Last, we show that the non-trivial primes $p$ for $K = \mathbb{Q}(i)$ and $K = \mathbb{Q}(\sqrt{-3})$ are characterized by modulo $p^2$ congruences involving Euler and Glaisher numbers respectively.