不可约尖点eta商的结构

中文摘要 AI 辅助

我们证明，水平为$N$的不可约尖点eta商由其在$O((\log_2 N)^{r-1})$个尖点处的消失阶唯一确定，其中$r$表示$N$的不同素因子个数。特别地，所有这类素幂水平的Fricke本征形式eta商，仅由其在$\infty$处的消失阶唯一确定。我们还证明，对所有$k\in\mathbb{N}$，存在权为$k/2$的本原不可约尖点eta商。由Zagier的权为$1/2$的本原全纯eta商列表的完备性可知，第1个Mersmann水平$\cl{M}_1=12$。本文证明，对所有$k\in\mathbb{N}$，第$k$个Mersmann水平可被$2^{2k}$整除。

英文摘要

We show that an ir{\-}reducible cuspidal eta quotient of level~$N$ is uniquely identified by its orders of vanishing at $O((\log_2 N)^{r-1})$ cusps, where $r$ denotes the number of distinct prime divisors of $N$. In particular, all such eta quotients which are Fricke eigenforms of a prime power level, are uniquely identified only by their orders of vanishing at~$\infty$. We also prove that for all $k\in\N$, there exists a primitive and irreducible cuspidal eta quotient of weight $k/2$. It follows from the exhaustiveness of Zagier's list of the primitive holomorphic eta quotients of weight~$1/2$ that the 1st Mersmann level, $\cl{M}_1=12$. Here we show that for all $k\in\N$, the $k$-th Mersmann level is divisible by $2^{2k}$.

发表机构

• Indian Institute of Science Education and Research Kolkata（印度科学教育研究学院加尔各答分校）

机构由 AI 辅助整理，请以论文原文为准。