根格$A_4$的相似子格
Similar sublattices of the root lattice $A_4$
摘要：
根据Conway、Rains和Sloane的结果，根格$A_4$的类似子格是可能的，对于每个索引是$m^2 + mn - n^2$形式的非零整数的平方。在这里，我们添加了一个建设性的方法，基于四元数代数$\mathbb{H} (\mathbb{Q} (\sqrt{5}))$的算术和第二类特定对合的存在性，这也提供了实际的子格和给定索引的不同解决方案的数量。相应的狄利克雷级数生成函数与二十世环的zeta函数密切相关。
英文摘要：
Similar sublattices of the root lattice $A_4$ are possible, according to a result of Conway, Rains and Sloane, for each index that is the square of a non-zero integer of the form $m^2 + mn - n^2$. Here, we add a constructive approach, based on the arithmetic of the quaternion algebra $\mathbb{H} (\mathbb{Q} (\sqrt{5}))$ and the existence of a particular involution of the second kind, which also provides the actual sublattices and the number of different solutions for a given index. The corresponding Dirichlet series generating function is closely related to the zeta function of the icosian ring.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
