Special values of Green's functions on hyperbolic $3$-space

TL;DR

This paper investigates special values of Green's functions on hyperbolic 3-space, demonstrating their averages relate to prime logs and units in real quadratic fields, and explores twisted averages yielding algebraic numbers.

Contribution

It extends the algebraicity results of Green's functions from modular curves to hyperbolic 3-space, providing explicit formulas for their averages and twisted averages.

Findings

Averages of Green's functions relate to logarithms of primes and units in real quadratic fields.

Twisted averages of Green's functions produce algebraic numbers.

The work generalizes known results from modular curves to hyperbolic 3-space.

Abstract

Gross, Kohnen and Zagier proved an averaged version of the algebraicity conjecture for special values of higher Green's functions on modular curves. In this work, we study an analogous problem for special values of Green's functions on hyperbolic $3$-space. We prove that their averages can be computed in terms of logarithms of primes and logarithms of units in real quadratic fields. Moreover, we study twisted averages of special values of Green's functions, which yield algebraic numbers instead of logarithms.