Observations on the "values" of the elliptic modular function j(tau) at real quadratics

TL;DR

This paper explores the values of the elliptic modular j-function at real quadratic irrationalities using Hecke's hyperbolic Fourier expansions, supported by numerical experiments.

Contribution

It introduces a new approach to defining and analyzing the values of j(tau) at real quadratic irrationals through hyperbolic Fourier expansions.

Findings

Numerical experiments reveal patterns in the values of j(tau) at real quadratic irrationals.

The approach provides insights into the behavior of modular functions at real quadratic points.

Abstract

We define "values" of the elliptic modular j-function at real quadratic irrationalities by using Hecke's hyperbolic Fourier expansions, and present some observations based on numerical experiments.