Unit Indices of Prime-Index Shanks Orders

Source abstract

For an integer $t\geq-1$, let $θ_t$ be the largest real root of $g_t(X)=X^3-tX^2-(t+3)X-1$, and let $R_t=\mathbb{Z}[θ_t]\subseteq\mathcal{O}_t=\mathcal{O}_{\mathbb{Q}(θ_t)}$. We prove that if the additive order index $q=[\mathcal{O}_t:R_t]$ is prime, then $[\mathcal{O}_t^\times:R_t^\times]=q$ exactly for $(t,q)=(3,3)$ and $(5,7)$, and the unit index is $1$ otherwise. The proof shows that $q$ has a unique prime $\mathfrak P$ above it and that $R_t=\mathbb{Z}+\mathfrak P^2$, thereby restricting the unit index to $1$ or $q$. It then invokes a maximal-order unit theorem in the wild branch and combines regulator and congruence bounds in the tame branch. We also determine the Picard kernel, the ideal class monoid, and every fiber of the associated integral matrix-conjugacy classification.