Gorenstein三次序的Yun ζ函数与超序ζ函数
Yun's zeta function and the overorder zeta function for Gorenstein cubic orders
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中文总结 AI 辅助
本文证明了Gorenstein三次序的Yun ζ函数与超序ζ函数的恒等式，给出了GL₃ Kloosterman狄利克雷级数的简短统一计算，替代了原有冗长的逐案枚举。
中文摘要 AI 辅助
对于每个Gorenstein三次ℤ-序，我们证明了一个恒等式，将Yun ζ函数（通过计数迹对偶的有限指数子模定义）与我们此前在关于GL₃的超越内窥镜工作中引入的显式超序ζ函数等同起来。该恒等式被Deng-Espinosa和Lee的早期草稿提出为猜想A，Lee随后证明了超序ζ函数的函数方程，使Deng-Espinosa对平凡表示的孤立完全无条件，其论证明确计算了局部因子。我们的证明独立于Lee的证明，且不单独计算三次超序因子：它匹配两边的自然分解并通过归纳法得出结论。作为应用，我们在Deng-Espinosa的泊松求和论证中给出了局部GL₃ Kloosterman狄利克雷级数的简短、统一计算。Deng-Espinosa中对Kloosterman级数的直接局部分析占了近90页，而我们的工作用简短的统一证明取代了其逐案枚举。
英文摘要
For every Gorenstein cubic $\mathbb Z$-order, we prove an identity equating Yun's zeta function, defined by counting finite-index submodules of the trace dual, with the explicit overorder zeta function introduced in our previous work on Beyond Endoscopy for $\mathrm{GL}_3$. This is posed as Conjecture A in an early draft of Deng-Espinosa and Lee subsequently proved the functional equation of the overorder zeta function, making the Deng--Espinosa isolation of the trivial representation fully unconditional. His argument computes the local factors explicitly. Our proof is independent of Lee's and does not evaluate the individual cubic overorder factors: it matches natural decompositions of the two sides and concludes by induction. As an application, we give a short, uniform evaluation of the local $\mathrm{GL}_3 $ Kloosterman Dirichlet series in the Poisson-summation argument of Deng--Espinosa. The direct local analysis of the Kloosterman series occupies nearly ninety pages in Deng--Espinosa while our treatment here replaces its case-by-case enumeration with a short uniform proof.
