# Values of modular functions at real quadratics and conjectures of Kaneko

## TL;DR

This paper proves some of Kaneko's conjectures regarding the values of modular functions at real quadratic irrationalities, which are defined via cycle integrals along geodesics, advancing understanding in this area.

## Contribution

It generalizes Kaneko's conjectures to a broader class of modular functions, providing new proofs and insights into their values at real quadratic points.

## Key findings

- Proved several of Kaneko's conjectures for general modular functions.
- Established explicit relations between modular function values and cycle integrals.
- Enhanced understanding of modular functions at real quadratic irrationalities.

## Abstract

In 2008, M. Kaneko made several interesting observations about the values of the modular j invariant at real quadratic irrationalities. The values of modular functions at real quadratics are defined in terms of their cycle integrals along the associated geodesics. In this paper we prove some of the conjectures of M. Kaneko for a general modular function.