In this paper we prove a conjecture of Farkas and Kra, which is a modular equation 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 conjecture holds for all odd integers $k \geq 3$. A new modular equation of Farkas and Kra type is also established.
"Proof of a Conjecture of Farkas and Kra." Taiwanese J. Math. 23 (6) 1317 - 1326, December, 2019. https://doi.org/10.11650/tjm/190301