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arXiv · 2608.23996

Geometric Realization of Finite Residue Casimirs and Weil Operators via Szeg\H{o} Kernels

Chuangqiang Hu·Lishan Yu
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Abstract

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 Szeg\H{o} 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.

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Chuangqiang Hu, Lishan Yu. 2026-08-25. Geometric Realization of Finite Residue Casimirs and Weil Operators via Szeg\H{o} Kernels. https://arxiv.org/abs/2608.23996

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