On Kronecker terms over global function fields

TL;DR

This paper develops a Kronecker limit formula over global function fields using Drinfeld period domains, leading to new analytic tools for studying special values of L-functions and heights of CM Drinfeld modules.

Contribution

It introduces a general Kronecker limit formula in the function field setting, connecting Drinfeld units, Colmez-type formulas, and L-function special values.

Findings

Established a Kronecker limit formula of arbitrary rank over global function fields.

Derived a Colmez-type formula for the stable Taguchi height of CM Drinfeld modules.

Applied the limit formula to special values of Rankin-Selberg and Godement-Jacquet L-functions.

Abstract

We establish a general Kronecker limit formula of arbitrary rank over global function fields with Drinfeld period domains playing the role of upper-half plane. The Drinfeld-Siegel units come up as equal characteristic modular forms replacing the classical $\Delta$. This leads to analytic means of deriving a Colmez-type formula for "stable Taguchi height" of CM Drinfeld modules having arbitrary rank. A Lerch-Type formula for "totally real" function fields is also obtained, with the Heegner cycle on the Bruhat-Tits buildings intervene. Also our limit formula is naturally applied to the special values of both the Rankin-Selberg $L$-functions and the Godement-Jacquet $L$-functions associated to automorphic cuspidal representations over global function fields.