通过Szegő核实现有限剩余卡西米尔与Weil算子的几何化
Geometric Realization of Finite Residue Casimirs and Weil Operators via Szegő Kernels
AI总结：
本文针对有限域上带无穷远点的光滑射影曲线，通过Szegő核实现有限剩余卡西米尔与Weil算子的几何化，为相关配对研究提供了几何基础。
AI中文摘要：
对于有限域上带有固定无穷远点的光滑射影曲线，我们建立了有限剩余对偶与几何核函数之间的通用对应关系。我们证明，有限剩余卡西米尔张量在几何上可实现为任意非零线丛的归一化Szegő核的固有主部，等价于该核作为有限剩余配对的再生核，与经典柯西积分公式完全类似。在多项式情形下，这些等价描述产生了包含秩2Weil算子的闭式公式，该算子被认定为经典差商，从而恢复了Hu-Ou的余项恒等式。这些结果为研究Anderson生成函数和Drinfeld模的Weil配对提供了几何基础。
英文摘要：
For a smooth projective curve over a finite field with a fixed point at infinity, we establish a universal correspondence between finite residue duality and geometric kernel functions. We prove that the finite residue Casimir tensor is realized geometrically as the intrinsic principal part of the normalized Szegő kernel for any acyclic line bundle, and equivalently that this kernel acts as a reproducing kernel for the finite residue pairing, in exact analogy with the classical Cauchy integral formula. In the polynomial case these equivalent descriptions yield a closed formula involving the rank-two Weil operator, identified with the classical divided difference, recovering a remainder identity of Hu--Ou. These results provide a geometric foundation for the study of Anderson generating functions and the Weil pairing for Drinfeld modules.
