There have been a number of recent works on the theory of period polynomials and their zeros. In particular, zeros of period polynomials have been shown to satisfy a "Riemann Hypothesis" in both classical settings and for cohomological versions extending the classical setting to the case of higher derivatives of $L$-functions. There thus appears to be a general phenomenon behind these phenomena. In this paper, we explore further generalizations by defining a natural analogue for Hilbert modular forms. We then prove that similar Riemann Hypotheses hold in this situation as well.

中文翻译：

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希尔伯特模形式周期多项式的黎曼假设

最近有许多关于周期多项式及其零点理论的著作。特别是，周期多项式的零点已被证明在经典设置和将经典设置扩展到 $L$ 函数的更高导数的情况下的上同调版本中都满足“黎曼假设”。因此，在这些现象背后似乎有一个普遍现象。在本文中，我们通过定义希尔伯特模形式的自然类似物来探索进一步的概括。然后我们证明类似的黎曼假设在这种情况下也成立。