Twisted traces of modular functions on hyperbolic $3$-space

TL;DR

This paper computes twisted traces of harmonic modular functions on hyperbolic 3-space, linking them to Fourier coefficients of the j-invariant and Dirichlet L-functions, revealing integrality properties and explicit formulas.

Contribution

It introduces explicit formulas for twisted traces of harmonic modular functions and Eisenstein series on hyperbolic 3-space, connecting them to classical number-theoretic functions.

Findings

Twisted traces of harmonic modular functions are integers.

Explicit formulas relate twisted traces to Fourier coefficients of the j-invariant.

Twisted traces of Eisenstein series are expressed via Dirichlet L-functions and divisor sums.

Abstract

We compute analogues of twisted traces of CM values of harmonic modular functions on hyperbolic $3$-space and show that they are essentially given by Fourier coefficients of the $j$-invariant. From this we deduce that the twisted traces of these harmonic modular functions are integers. Additionally, we compute the twisted traces of Eisenstein series on hyperbolic $3$-space in terms of Dirichlet $L$-functions and divisor sums.