The theory of elliptic modular forms has gained significant momentum from the discovery of relaxed yet well-behaved notions of modularity, such as mock modular forms, higher order modular forms, and iterated Eichler–Shimura integrals. Applications beyond number theory range from combinatorics, geometry, and representation theory to string theory and conformal field theory. We unify these relaxed notions in the framework of vector-valued modular forms by introducing a new class of $\mathrm{SL}_2 (\mathbb{Z})$-representations: virtually real-arithmetic types. The key point of the paper is that virtually real-arithmetic types are in general not completely reducible. We obtain a rationality result for Fourier and Taylor coefficients of associated modular forms.

中文翻译：

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几乎是实数算术类型I的模块形式：混合模拟模块形式产生矢量值的模块形式

椭圆模块化形式的理论已从发现宽松但表现良好的模块化概念（例如模拟模块化形式，高阶模块化形式和迭代的Eichler-Shimura积分）中获得了巨大动力。数论之外的应用范围从组合论，几何学和表示论到弦论和共形场论。通过引入新的$ \ mathrm {SL} _2（\ mathbb {Z}）$表示形式：虚拟实数类型，我们在向量值模块化形式的框架中统一了这些宽松的概念。本文的重点是，实际上实数运算类型通常不能完全还原。我们获得了相关模块形式的傅里叶和泰勒系数的合理性结果。