发表机构
Kurume Institute of Technology; Osaka University of Health and Sport Science（久留米工业大学; 大阪健康运动科学大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文用代数方法（基于主同余子群连接矩阵与伽罗瓦理论）证明Kaneko-Zagier微分方程在特定权重下无模形式解，并完整分类了解空间的维数。
AI 中文摘要
本文给出了权重$k$的Kaneko-Zagier微分方程存在模形式解的条件的一个代数证明。与先前依赖$SL_2(\mathbb{Z})$的表示论方法不同，我们的方法利用主同余子群$\Gamma(N)$的连接矩阵。通过从表示矩阵的交换子推导出同时三角化的条件，并应用伽罗瓦理论，我们证明了当分数$(k+1)/6 = n/m$的分母$m$满足$m=1$或$m \ge 7$时，$\Gamma(N)$的关联二维表示是不可约的。因此，我们确定了不存在模形式解的权重，并获得了此类解空间的维数的完整分类。
英文摘要
In this paper, we provide an algebraic proof of the condition for the existence of modular form solutions to the Kaneko-Zagier differential equation of weight $k$. Unlike previous representation-theoretic approaches relying on $SL_2(\mathbb{Z})$, our method employs the connection matrices of the principal congruence subgroup $Γ(N)$. By deriving a condition for simultaneous triangularizability from the commutators of the representation matrices and applying Galois theory, we prove that the associated two-dimensional representation of $Γ(N)$ is irreducible when the denominator $m$ of the fraction $(k+1)/6 = n/m$ satisfies $m=1$ or $m \ge 7$. Consequently, we identify the weights that admit no modular form solutions and obtain a complete classification of the dimensions of the spaces of such solutions.
