论文标题
一致性的级别$ 1 $ cusp形式的半融合体重
Congruences for Level $1$ cusp forms of half-integral weight
论文作者
论文摘要
假设$ \ ell \ geq 5 $是Prime。对于一个正整数$ n $，$ 4 \中n $，以前的作品研究了$γ_0（n）$的半综合重量模块化表格的属性，这些属性在有限的许多Square类Modulo $ \ ell $上得到了支持，在某些情况下，在某些情况下证明了这些形式与单个可变的Theta theta系列的形象一致，这些形式是theta seriver的单个theta serivers the the the ramanus $ ramanuy的图像。在这里，我们研究了$ \ operatatorName {sl} _ {2}（\ Mathbb {z}）$上半融合权重的模块化形式的类似问题。令$η$为Dedekind ETA功能。对于各种重量，我们证明，在$ \ operatorname {sl} _ {2}（\ Mathbb {z}）$上，每一个半综合的权重模块化形式都可以在许多方形类中支持，以$η^$η^$ feriv n od modulo $ \ ell可以写下。
Suppose that $\ell \geq 5$ is prime. For a positive integer $N$ with $4 \mid N$, previous works studied properties of half-integral weight modular forms on $Γ_0(N)$ which are supported on finitely many square classes modulo $\ell$, in some cases proving that these forms are congruent to the image of a single variable theta series under some number of iterations of the Ramanujan $Θ$-operator. Here, we study the analogous problem for modular forms of half-integral weight on $\operatorname{SL}_{2}(\mathbb{Z})$. Let $η$ be the Dedekind eta function. For a wide range of weights, we prove that every half-integral weight modular form on $\operatorname{SL}_{2}(\mathbb{Z})$ which is supported on finitely many square classes modulo $\ell$ can be written modulo $\ell$ in terms of $η^{\ell}$ and an iterated derivative of $η$.
