Several-variable Kronecker limit formula over global function fields
Fu-Tsun Wei
TL;DR
This work extends Kronecker limit formulas to global function fields in a several-variable setting by leveraging the Berkovich Drinfeld period domains. The authors derive integral representations that express derivatives of non-holomorphic and Jacobi-type Eisenstein series as averaged period integrals over Heegner cycles, and they provide period interpretations of the Kronecker terms for Dedekind zeta and Dirichlet L-functions. The key innovations include a robust Berkovich interpretation, a theta-theta approach to products of norms, and explicit connections to Drinfeld discriminants, Klein forms, and Drinfeld--Siegel units. These results yield a unified, geometric understanding of special values of automorphic L-functions in the function-field setting and pave the way for automorphic applications via base change and induction in future work.
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.