极值拟模形式系数的渐近展开及Kaneko和Koike猜想
Asymptotic expansions for the coefficients of extremal quasimodular forms and a conjecture of Kaneko and Koike
摘要：
极值拟模形式是M.~ Kaneko和M.Koike等给出了在$i\infty$处具有最大可能消失阶的拟模形式.我们给出了这种形式的傅里叶系数的渐近公式。然后用这个公式证明了这种深度形式$\leq4$的傅里叶系数除了100%以外都是正的，这部分地解决了M.~题名其余部分：M.基于构造性估计的数值实验证实了关于权值$\leq200$和深度介于$1$和$4$之间的猜想。
英文摘要：
Extremal quasimodular forms have been introduced by M.~Kaneko and M.Koike as as quasimodular forms which have maximal possible order of vanishing at $i\infty$. We show an asymptotic formula for the Fourier coefficients of such forms. This formula is then used to show that all but finitely many Fourier coefficients of such forms of depth $\leq4$ are positive, which partially solves a conjecture stated by M.~Kaneko and M.Koike. Numerical experiments based on constructive estimates confirm the conjecture for weights $\leq200$ and depths between $1$ and $4$.
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