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