Multidimensional small divisor functions

TL;DR

This paper generalizes the construction of small divisor functions to multi-indices, producing explicit examples of polar harmonic Maa{ ext} forms in various even dimensions, expanding the understanding of these functions in higher-dimensional settings.

Contribution

It extends previous work to multi-indices and provides explicit examples of polar harmonic Maa{ ext} forms in multiple even dimensions.

Findings

Explicit examples in dimensions 4, 6, 8, and 10.

Construction of polar harmonic Maa{ ext} forms of non-positive integral weight.

Generalization of small divisor functions to multi-indices.

Abstract

This is a short note generalizing the construction from arXiv:1906.07410, arXiv:2009.04955 to multi-indices. We recommend to consider both references first. We obtain polar harmonic Maa{\ss} forms of non-positive integral weight if the dimension is even and greater than $2$. We provide explicit examples in dimension $4$, $6$, $8$, and $10$.