We complete several generating functions to non-holomorphic modular forms in two variables. For instance, we consider the generating function of a natural family of meromorphic modular forms of weight two. We then show that this generating series can be completed to a smooth, non-holomorphic modular form of weights 3/2 and two. Moreover, it turns out that the same function is also a modular completion of the generating function of weakly holomorphic modular forms of weight 3/2, which prominently appear in work of Zagier on traces of singular moduli.

中文翻译：

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关于某些生成函数的模块化完成

我们对两个变量中的非全纯模形式完成了几个生成函数。例如，我们考虑权重二的亚纯模形式的自然族的生成函数。然后我们证明这个生成序列可以完成为一个平滑的、非全纯的权重为 3/2 和 2 的模形式。此外，事实证明，相同的函数也是权重 3/2 的弱全纯模形式的生成函数的模完成，这在 Zagier 关于奇异模迹的工作中突出地出现。