Hilbert-Asai Eisenstein series, regularized products, and heat kernels

Abstract

In a famous paper, Asai indicated how to develop a theory of Eisenstein series for arbitrary number fields, using hyperbolic 3-space to take care of the complex places. Unfortunately he limited himself to class number 1. The present paper gives a detailed exposition of the general case, to be used for many applications. First, it is shown that the Eisenstein series satisfy the authors' definition of regularized products satisfying the generalized Lerch formula, and the basic axioms which allow the systematic development of the authors' theory, including the Cramér theorem. It is indicated how previous results of Efrat and Zograf for the strict Hilbert modular case extend to arbitrary number fields, for instance a spectral decomposition of the heat kernel periodized with respect to ${\rm SL}_2$ of the integers of the number field. This gives rise to a theta inversion formula, to which the authors' Gauss transform can be applied. In addition, the Eisenstein series can be twisted with the heat kernel, thus encoding an infinite amount of spectral information in one item coming from heat Eisenstein series. The main expected spectral formula is stated, but a complete exposition would require a substantial amount of space, and is currently under consideration.