Classical and adelic Eisenstein series

TL;DR

This paper explores the relationship between classical and adelic Eisenstein series through explicit calculations, revealing the automorphic representations they generate, especially for weight 2, and extending to series with level and character.

Contribution

It provides explicit calculations connecting classical and adelic Eisenstein series and characterizes the automorphic representations they generate, including the complex case of weight 2.

Findings

Determined automorphic representations generated by Eisenstein series.

Established explicit links between classical and adelic Eisenstein series.

Analyzed Eisenstein series of weight 2 with level and character.

Abstract

We carry out "Hecke summation" for the classical Eisenstein series $E_k$ in an adelic setting. The connection between classical and adelic functions is made by explicit calculations of local and global intertwining operators and Whittaker functions. In the process we determine the automorphic representations generated by the $E_k$, in particular for $k=2$, where the representation is neither a pure tensor nor has finite length. We also consider Eisenstein series of weight $2$ with level, and Eisenstein series with character.