Distributions de s\'eries d'Eisenstein presque holomorphes sur un corps totalement r\'eel

TL;DR

This paper defines and analyzes nearly-holomorphic Eisenstein series over totally real fields, demonstrating their properties, computing Fourier coefficients, and connecting them to classical Eisenstein series in simple cases.

Contribution

It introduces a new class of Eisenstein series as distributions on totally real fields, with explicit properties and Fourier coefficient calculations, linking abstract and classical forms.

Findings

Eisenstein series are expressed as distributions with automorphic properties

Fourier coefficients are explicitly computed for these series

In simple cases, the series reduce to classical Eisenstein series

Abstract

At first a type of Eisenstein series is defined as distributions giving nearly-holomorphic automorphic forms on a totally real field, with different expressions (integral, summation) ; then these are shown to satisfied the expected properties, and the Fourier coefficients are explicitly computed. Finally, an explicit computation shows that in the simplest case, the abstract object considered simplify down to the most classical version of Eisenstein series.