论文标题
Hecke组模块化形式的多项式插值
Polynomial interpolation of modular forms for Hecke groups
论文作者
论文摘要
延长J. Raleigh的工作，我们计算多项式$ p_ {n，f}（x）$与某些家庭相关的$ f = \ {f_m \} _ {m = 3，4，...} $ for Hecke groups $ g（λ_m）的模块化形式的$ g（λ_m）$与P_ { $ f_m $的扩展。我们以众所周知的Hauptmoduln的傅立叶扩展或某些分数和某些分数表达$ p_ {n，f} $。通过研究$ P_N $的复杂根，我们将它们与Lehmer关于Ramanujan的Tau功能的问题联系起来。我们回顾了三角函数的理论和Hecke的模块化形式理论，以建立我们的代码的基础，其中一些起源于J. Leo的论文。本文是数值实验的说明。其中唯一的定理属于其他人的工作，我们如上所述进行了审查。
Extending work of J. Raleigh, we compute polynomials $P_{n,F}(x)$ associated to certain families $F = \{f_m\}_{m = 3, 4, ...}$ of modular forms for Hecke groups $G(λ_m)$ with the property that $P_{n,F}(m)$ is the $n^{th}$ coefficient in the Fourier expansion of $f_m$. We express the $P_{n,F}$ in terms of the Fourier expansions of well-known Hauptmoduln, or in terms of certain divisor-sums. By studying the complex roots of the $P_n$, we relate them to Lehmer's question about Ramanujan's tau function. We review the theory of triangle functions and Hecke's theory of modular forms in order to establish a basis for our code, some of which originates in the dissertation of J. Leo. The article is an account of numerical experiments; the only theorems in it belong to work by others that we review as described above.
