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

In a famous paper, Asai indicated how to develop a theory of Eisenstein ser ies for arbitrary number fields, using hyperbolic 3-space to take care of t he complex places. Unfortunately he limited himself to class number 1. The present paper gives a detailed exposition of the general case, to be used f or 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 Cramer theorem. It is indicated how previous results of Efrat and Zograf for the strict Hilbert modular case ex tend to arbitrary number fields, for instance a spectral decomposition of t he heat kernel periodized with respect to SL2 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 expect ed spectral formula is stated, but a complete exposition would require a su bstantial amount of space, and is currently under consideration.