# Proof of a conjecture of Farkas and Kra

## TL;DR

This paper proves a conjectured modular equation by Farkas and Kra involving modular forms for certain congruence subgroups, establishing its validity for all odd integers greater than or equal to 2, and introduces a new related modular equation.

## Contribution

The paper confirms a longstanding conjecture of Farkas and Kra for all odd integers and introduces a novel modular equation of similar type.

## Key findings

- Confirmed the conjectured modular equation for all odd k ≥ 2
- Established a new modular equation of Farkas and Kra type
- Extended understanding of modular forms for congruence subgroups

## Abstract

In this paper we prove a conjectured modular equation of Farkas and Kra, which involving a half sum of certain modular form of weight $1$ for congruence subgroup $\Gamma_1(k)$ with any prime $k$. We prove that their conjectured identity holds for all odd integer $k\ge 2$. A new modular equation of Farkas and Kra type is also established.