论文标题
在kaneko和Koike的猜想中
On a conjecture of Kaneko and Koike
论文作者
论文摘要
2006年，Kaneko和Koike定义了极端的绝模型形式，并证明了它们的存在深度为$ 1 $和$ 2 $。在最多$ 4 $的最多限制并限制了深度的情况之后，他们猜想了某种形式的傅立叶系数的某种结合。更确切地说，系数分母的主要因素要求小于重量。最近，Pellarin证明了这一猜想，如果深度为$ 1 $，权重为6美元。在本文中，我们将图片深入完成$ 1 $。首先，我们表明他的结果意味着在每个整数$ k \ geq 0 $的重量$ 6K+4 $的情况下，都意味着相同的结果。其次，在重量$ W = 6K $的情况下，我们将其证明的策略调整为重量$ W = 6K+2 $的最后一个情况。最后，我们为他和我们的中级结果提供了所有计算细节，因为这些细节对于他的证明至关重要，但在他在Arxiv中的博览会中省略了：1910.11668。彼得·格拉格纳（Peter Grabner）平行且独立于这项工作，证明了上述猜想的一般性，请参见Arxiv：2002.02736。
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.
