素指数Shanks序的单位指数

中文摘要 AI 辅助

对于整数t≥-1，设θ_t是g_t(X)=X³-tX²-(t+3)X-1的最大实根，R_t=ℤ[θ_t]包含于O_t=ℚ(θ_t)的整数环O_ℚ(θ_t)。我们证明：若加法序指数q=[O_t:R_t]为素数，则仅当(t,q)=(3,3)和(5,7)时，单位群指数[O_t^×:R_t^×]=q，其余情况单位指数为1。证明显示q上方存在唯一素理想𝔓，且R_t=ℤ+𝔓²，由此将单位指数限制为1或q；再结合非分歧分支的极大序单位定理与分歧分支的 regulator（ regulator 译为 regulator，此处保留原名）及同余界。我们还确定了皮卡核、理想类幺半群，以及相关整矩阵共轭分类的所有纤维。

英文摘要

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 set $R_t=\mathbb{Z}[θ_t]\subseteq\mathcal{O}_t=\mathcal{O}_{\mathbb{Q}(θ_t)}$, $N_t=[\mathcal{O}_t:R_t]$, and $\varepsilon_t=[\mathcal{O}_t^\times:R_t^\times]$. We determine $\varepsilon_t$ when $N_t$ is squarefree, $27$, or $343$: the only nontrivial indices in these cases are $\varepsilon_3=3$, $\varepsilon_5=7$, $\varepsilon_{12}=13$, and $\varepsilon_{54}=19$. Local conductor calculations give $\varepsilon_t\mid N_t$ for squarefree $N_t$. For arbitrary $N_t$, a regulator comparison shows that $t\geq2N_t$ implies $\varepsilon_t=1$. When $N_t=p^3$ with $p\neq3$ a rational prime, this bound and the index criterion leave at most four possible parameters with $\varepsilon_t>1$ for each fixed $p$. For arbitrary additive index, we also prove that $13\mid\varepsilon_t$ 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 $\operatorname{Pic}(\mathcal{O}_t)$, and the corresponding integral matrix-conjugacy classes.

发表机构

• Yang-En University（杨恩大学）

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