论文标题
模块化网格的算术
The arithmetic of modular grids
论文作者
论文摘要
模块化网格是一对序列$（f_m）_m $和$（g_n）_n $弱塑形模块化形式，以至于几乎所有$ m $ and $ n $，$ q^n $ in $ f_m $ in $ q^n $ in $ q^m $ in $ q^m $ in $ g_n $中的nate $ q^n $。 Zagier证明了这种系数双重性的重量$ 1/2 $和Kohnen Plus Space的$ 3/2 $，并且在Poincaré系列中出现了此类网格，用于模块化的整体权重形式以及许多其他情况。对于每个组$γ\ subseteq {\ subseteq {\ text {sl}} _ 2（\ Mathbb {r}）$ Conconsurable，我们提供了弱塑形模块化形式或半积分重量的弱塑形模块化形式或半融合重量的空间的二元组的一般证明。 $ {\ text {sl}} _ 2（\ mathbb {z}）$。我们构建双变量生成的函数，编码这些模块化形式，并在所得的模块化网格上研究线性操作。
A modular grid is a pair of sequences $(f_m)_m$ and $(g_n)_n$ of weakly holomorphic modular forms such that for almost all $m$ and $n$, the coefficient of $q^n$ in $f_m$ is the negative of the coefficient of $q^m$ in $g_n$. Zagier proved this coefficient duality in weights $1/2$ and $3/2$ in the Kohnen plus space, and such grids have appeared for Poincaré series, for modular forms of integral weight, and in many other situations. We give a general proof of coefficient duality for canonical row-reduced bases of spaces of weakly holomorphic modular forms of integral or half-integral weight for every group $Γ\subseteq {\text{SL}}_2(\mathbb{R})$ commensurable with ${\text{SL}}_2(\mathbb{Z})$. We construct bivariate generate functions that encode these modular forms, and study linear operations on the resulting modular grids.
