Using Continued Fractions to Compute Iwasawa Lambda Invariants of Imaginary Quadratic Number Fields

Abstract

Let $\ell >3$ be a prime such that $\ell \equiv 3(\mathrm{mod}4)$ and $\mathbb{Q}(\sqrt{\ell })$ has class number 1. Then Hirzebruch and Zagier noticed that the class number of $\mathbb{Q}(\sqrt{-\ell })$ can be expressed as $h(-\ell )=(1/3)(b_1+b_2+\cdots +b_m)-m$ where the $b_i$ are partial quotients in the `minus' continued fraction expansion $\sqrt{\ell }=[[b_0;\overline{b_1,b_2,\dots ,b_m}]]$. For an odd prime $p\ne \ell$, we prove an analogous formula using these $b_i$ which computes the sum of Iwasawa lambda invariants ${\lambda }_p(-\ell )+{\lambda }_p(-4)$ of $\mathbb{Q}(\sqrt{-\ell })$ and $\mathbb{Q}(\sqrt{-1})$. In the case that $p$ is inert in $\mathbb{Q}(\sqrt{-\ell })$, the formula pleasantly simplifies under some additional technical assumptions.

18 pages