# Vector-valued Eisenstein series of small weight

## TL;DR

This paper investigates vector-valued Eisenstein series of small weights for the Weil representation, analyzing their transformation properties and revealing arithmetic information in their coefficients.

## Contribution

It provides a detailed description of the transformation laws and explores arithmetic properties of coefficients for small weight Eisenstein series.

## Key findings

- Coefficients encode interesting arithmetic data
- Transformation laws are explicitly described
- Examples illustrate the arithmetic significance

## Abstract

We study the (mock) Eisenstein series $E_k$ of weight $k \in \{1,3/2,2\}$ for the Weil representation on an even lattice, defined as the result of Bruinier and Kuss's coefficient formula for the Eisenstein series naively evaluated at $k$. We describe the transformation law of $E_k$ in general. Most of this note is dedicated to collecting examples where the coefficients of $E_k$ contain interesting arithmetic information. Finally we make a few remarks about the case $k=1/2$.