On a conjecture of Kaneko and Koike

Abstract

In 2006, Kaneko and Koike defined extremal quasimodular forms and proved their existence in depth $1$ and $2$. After normalizing and restricting to the case of depth at most $4$, they conjectured a certain bound on the Fourier coefficients of such forms. More precisely, the prime factors of the denominators of the coefficients are requested to be smaller than the weight. Recently, Pellarin proved this conjecture in the case of depth $1$ and weight divisible by $6$. In this paper, we complete the picture in depth $1$. First, we show that his result implies the same result in the case of weight $6k+4$ for every integer $k \geq 0$ directly. Secondly, we adapt the strategy of his proof in the case of weight $w = 6k$ to the last case of weight $w = 6k+2$. Finally, we provide all computational details to both his and our intermediate results, since those details are essential to his proof, but were omitted during his exposition in arXiv:1910.11668. Parallel and independent from this work, Peter Grabner proved the aforementioned conjecture in full generality, see arXiv:2002.02736.