Several-variable Kronecker limit formula over global function fields

Abstract

We establish Kronecker-type first and second limit formulas for "non-holomorphic" and "Jacobi-type" Eisenstein series over global function fields in the several-variable setting. Our main theorem demonstrates that the derivatives of these Eisenstein series can be understood as averaged integrals of certain period quantities along the associated "Heegner cycles" on Drinfeld modular varieties. A key innovation lies in our use of the Berkovich analytic structure of the Drinfeld period domains, which enables the parametrization of the Heegner cycles in question by Euclidean "parallelepiped" regions. This approach also facilitates a unified and streamlined formulation and proof of our results. Finally, we apply these formulas to provide period interpretations of the "Kronecker terms" of Dedekind-Weil zeta functions and Dirichlet $L$-functions associated with ring and ray class characters.