AN ANALYTIC PERSPECTIVE OF WEIL RECIPROCITY

Abstract

In Cogdell et al., LMS Lecture Notes Series 459, 393–427 (2020), the authors proved a type of Kronecker’s limit formula associated to any divisor $D$ on any smooth Kähler manifold $X$, assuming that $D$ is smooth in codimension one. In the present article, it is shown how the aforementioned analogue of Kronecker’s limit formula applies to reprove and generalize Weil reciprocity. More precisely, we extend Weil reciprocity to (suitably normalized) meromorphic modular forms of even weight on a smooth, compact Riemann surface, and present a variant of Weil reciprocity for a class of harmonic functions with special types of singularities on a finite volume quotient of a symmetric space or a compact, smooth projective Kähler variety. We also prove an integral version of Weil reciprocity for a compact, smooth projective Kähler variety.