Unit Indices of Shanks Orders

Abstract

For an integer $t\ge -1$, let ${\theta }_t$ be the largest real root of $g_t(X)=X^3-t{X}^2-(t+3)X-1$, and set $R_t=\mathbb{Z}[{\theta }_t]\subseteq {\mathcal{O}}_t={\mathcal{O}}_{\mathbb{Q}({\theta }_t)}$, $N_t=[{\mathcal{O}}_t:R_t]$, and ${\epsilon }_t=[{\mathcal{O}}_t^{\times }:R_t^{\times }]$. We determine ${\epsilon }_t$ when $N_t$ is squarefree, $27$, or $343$: the only nontrivial indices in these cases are ${\epsilon }_3=3$, ${\epsilon }_5=7$, ${\epsilon }_{12}=13$, and ${\epsilon }_{54}=19$. Local conductor calculations give ${\epsilon }_t\mid N_t$ for squarefree $N_t$. For arbitrary $N_t$, a regulator comparison shows that $t\ge 2N_t$ implies ${\epsilon }_t=1$. When $N_t=p^3$ with $p\ne 3$ a rational prime, this bound and the index criterion leave at most four possible parameters with ${\epsilon }_t>1$ for each fixed $p$. For arbitrary additive index, we also prove that $13\mid {\epsilon }_t$ if and only if $t=12$ or $66$, with unit index $13$ in both cases. The proof determines the rational solutions of a plane quartic equation by an explicit genus-two descent and a local Chabauty argument, proving Louboutin's Conjecture 19 on its integral solutions. When $N_t$ is a rational prime, we determine the Picard kernel, the cardinality and fibers of the ideal class monoid over $\mathrm{Pic}({\mathcal{O}}_t)$, and the corresponding integral matrix-conjugacy classes.