49阶初等阿贝尔群的整数群行列式
Integer group determinants for the elementary abelian group of order 49
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中文总结 AI 辅助
本文通过分类能被7整除的取值，完整确定了$C_7\times C_7$的整数群行列式集合，给出赋值10和11的余因子条件，并求得最小正取值。
中文摘要 AI 辅助
设$G=C_7\times C_7$，并令$S(G)$表示其群行列式的整数取值集合。我们通过分类能被7整除的取值来确定$S(G)$；互素的取值已知。每个非零的可被7整除的取值其7-adic赋值至少为10，且所有$7^{12}$的倍数都会出现。在赋值10和11处，我们给出了关于$\mathbb{Z}[\zeta_7]$中理想的加性不变量的余因子满足的充分必要条件。第一个条件是素理想不变量的有界符号和。第二个条件要求存在一个非零不变对的素理想，其范数整除余因子。两个余因子集合都不是模任意正整数的同余类之并。最小的正可被7整除的取值为$43\cdot7^{10}$，而赋值11的最小正取值为$8\cdot7^{11}$。证明结合了从特征值进行整数重构与对全局单位模7的像的计算。最后，我们指出在素数至少为11时出现的额外局部条件和实现问题。
英文摘要
Let $G=C_7\times C_7$, and let $S(G)$ denote the set of integer values of its group determinant. We determine $S(G)$ by classifying the values divisible by $7$; the coprime values are already known. Every nonzero divisible value has valuation at least $10$, and every multiple of $7^{12}$ occurs. At valuations $10$ and $11$, we give necessary and sufficient conditions on the cofactor in terms of two additive invariants of ideals in $\mathbb{Z}[ζ_7]$. The first condition is a bounded signed sum of prime-ideal invariants. The second requires a prime ideal with nonzero invariant pair whose norm divides the cofactor. Neither cofactor set is a union of congruence classes modulo any positive integer. The least positive divisible value is $43\cdot7^{10}$, and the least positive value of valuation $11$ is $8\cdot7^{11}$. The proof combines integral reconstruction from character values with a calculation of the global-unit image modulo $7$. We conclude by identifying the additional local conditions and realization problems that arise at primes at least $11$.
发表机构
- Mahidol University International College（玛希隆大学国际学院）
机构由 AI 辅助整理，请以论文原文为准。
