AI 中文总结
该研究针对有限域上曲线的平方零扩张，推广了PGL₂的自守函数相关结果，证明了PGL₃下某猜想的新案例，还确定了对应球面尖与赫克有限函数的最优支撑界。
AI 中文摘要
我们研究有限域上曲线$\boldsymbol{\bar{C}}$的平方零扩张$C$的自守函数，该研究由Braverman-Kazhdan-Polishchuk在文献[BKP23]中首次发起。更确切地说，我们研究分裂连通约化群$G$下，上述文献中引入的轨道分解函数的尖性与赫克有限性，推广了$G=\boldsymbol{\text{PGL}}_2$的部分结果。由此，对于$G=\boldsymbol{\text{PGL}}_3$，我们证明了文献[BK23]中关于非分歧赫克有限函数空间有限维性猜想的一个新案例。我们还根据约化曲线上$G$-丛的Harder-Narasimhan层，给出了球面尖函数与赫克有限函数的支撑界的表述。结合表示论构造及其在$C$上的$G$-丛、$\boldsymbol{\bar{C}}$上的特定扭曲$G$-希格斯丛的几何解释，我们在若干情形下计算了最优界，特别确定了$G=\boldsymbol{\text{PGL}}_3$时的最优界。
英文摘要
We study automorphic functions for square-zero extensions $C$ of curves $\overline{C}$ over finite fields, a study initiated by Braverman-Kazhdan-Polishchuk in [BKP23]. More precisely, we study the cuspidality and Hecke-finiteness of the functions in the orbit decomposition introduced in loc. cit. for split connected reductive groups $G$, generalizing some of the results for $G=\mathrm{PGL}_2$. As a result, for $G=\mathrm{PGL}_3$, we prove a new case of a conjecture in [BK23] concerning the finite-dimensionality of the space of unramified Hecke-finite functions. We also introduce a formulation of support bounds for spherical cuspidal and Hecke-finite functions in terms of the Harder-Narasimhan stratification of $G$-bundles on the reduced curve. Using representation-theoretic constructions together with their geometric interpretations in terms of $G$-bundles on $C$ and certain twisted $G$-Higgs bundles on $\overline{C}$, we compute the optimal bounds in several cases and, in particular, determine the optimal bound for $G=\mathrm{PGL}_3$.
Comments50 pages
