二次型的最小阻碍模
The minimal obstruction modulus for quadratic forms
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中文总结 AI 辅助
本文研究本原正定整二元二次型的最小阻碍模，分两种判别式情形完全确定其值，并拓展得到本原三元对角型的对应不变量。
中文摘要 AI 辅助
设$Q = ax^2+bxy+cy^2$是本原正定整二元二次型，其判别式为$Δ= b^2-4ac$。已知$Q$存在局部阻碍，即存在整数$k,l$使得$Q \not\equiv l \pmod k$。我们研究最小阻碍模$κ_Q := \{k \in \mathbb{Z}_{\geq 1} \mid \text{存在} \\\\ l \text{使得} \\\\ Q \not\equiv l \pmod k \}$的最小值，分$Δ\equiv 0 \pmod 4$和$Δ\equiv 1 \pmod 4$两种情况完全确定了$κ_Q$的取值。我们还确定了本原三元对角型的类似不变量。
英文摘要
Let $Q = ax^2+bxy+cy^2$ be a primitive positive definite integral binary quadratic form with discriminant $Δ= b^2-4ac$. It is known that $Q$ admits a local obstruction; that is, there exist $k,l \in \mathbb{Z}$ such that $Q \not\equiv l \pmod k$. We study the minimal obstruction modulus $κ_Q := \min \{k \in \mathbb{Z}_{\geq 1} \mid \text{there exists } l \text{ such that } Q \not\equiv l \pmod k \}$, and we determine $κ_Q$ completely, treating the cases $Δ\equiv 0 \pmod 4$ and $Δ\equiv 1 \pmod 4$ separately. We also determine the analogous invariants for primitive ternary diagonal forms.
