负指标 Poincaré 级数的单项式

中文摘要 AI 辅助

众所周知，Eisenstein 级数 $E_k$ 不能写成两个较低权重的 Eisenstein 级数的乘积，除了 $E_{14}=E_4 E_{10}= E_4^2 E_6= E_6 E_8$。类似的结果对于 Hecke 特征形式也是已知的。同样已知，Eisenstein 级数的单项式之间的等式可以归结为上述恒等式。最近，本文作者与 E. Saha 一起研究了指标为 $1$ 的 Poincaré 尖形式的可能的单项式关系。到目前为止，研究单项式关系的问题一直局限于全纯模形式的框架。对于偶整数 $k\ge 4$ 和负整数 $m$，权重为 $k$、指标为 $m$ 的 Poincaré 级数 $G_{k}(z,m)$ 是一个弱全纯模形式，在 $ι\infty$ 处有一个阶为 $-m$ 的极点。显然，对于 $m<0$，Poincaré 级数 $G_k(z,m)$ 不能写成两个相同指标 $m$ 的较低权重 Poincaré 级数的乘积。在本文中，我们研究了对于任意 $m<0$，Poincaré 级数 $G_k(z,m)$ 的单项式之间可能的等式。特别地，我们证明了对于任何满足 $0.06\le\{-2mπ\}\le0.99$ 的 $m<0$，由权重 $k\ge 50$ 的 $G_k(z,m)$ 组成的两个单项式永远不会相等，其中 $\{x\}$ 表示实数 $x$ 的小数部分。根据 Weyl 的等分布准则，至少 $93\\%$ 的 $m<0$ 满足上述不等式。

英文摘要

It is well known that an Eisenstein series $E_k$ cannot be written as a product of two lower-weight Eisenstein series, except for $E_{14}=E_4 E_{10}= E_4^2 E_6= E_6 E_8$. A similar result is also known for the Hecke eigenforms. It is also known that equalities among the monomials of the Eisenstein series can be reduced to the above-mentioned identities. Recently, the present author, along with E. Saha studied the possible monomial relations of the Poincaré cusp forms of index $1$. So far, the problem of studying the monomial relations has been restricted to the setup of holomorphic modular forms. For an even integer $k\ge 4$ and a negative integer $m$, the Poincaré series $G_{k}(z,m)$ of weight $k$ and index $m$ is a weakly holomorphic modular form having a pole of order $-m$ at $ι\infty$. It is immediate that the Poincaré series $G_k(z,m)$ for $m<0$ can not be written as a product of two lower-weight Poincaré series of the same index $m$. In this article, we investigate the possible equalities among the monomials of the Poincaré series $G_k(z,m)$ for arbitrary $m<0$. In particular, we show that for any $m<0$ such that $0.06\le\{-2mπ\}\le0.99$, two monomials composed of $G_k(z,m)$ of the weights $k\ge 50$ are never equal, where $\{x\}$ denotes the fractional part of a real number $x$. In view of Weyl's equidistribution criterion, at least $93\%$ of $m<0$ satisfies the above inequality.

发表机构

• Institute of Mathematical Sciences（马修研究所）

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