In recent years, a number of papers have been devoted to the study of zeros of period polynomials of modular forms. In the present paper, we study cohomological analogues of the Eichler–Shimura period polynomials corresponding to higher L-derivatives. We state a general conjecture about the locations of the zeros of the full and odd parts of the polynomials, in analogy with the existing literature on period polynomials, and we also give numerical evidence that similar results hold for our higher derivative “period polynomials” in the case of cusp forms. The unimodularity of the roots seems to be a very subtle property which is special to our “period polynomials”. This is suggested by numerical experiments on families of perturbed “period polynomials” (Section ) suggested by Zagier. We prove a special case of our conjecture in the case of Eisenstein series.

中文翻译：

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Eichler同调和与L函数导数相关的多项式的零

近年来，许多论文致力于模块化形式的周期多项式的零点的研究。在本文中，我们研究了对应于较高L的Eichler-Shimura时期多项式的同调类似物-衍生物。与关于周期多项式的现有文献相类似，我们对多项式的全部和奇数部分的零的位置进行了一般性的猜想，并且我们还提供了数值证据，证明了类似的结果适用于我们的更高阶导数“周期多项式”。尖头形式。根的单模性似乎是一个非常微妙的特性，这对于我们的“周期多项式”来说是特殊的。Zagier建议的扰动“周期多项式”族（部分）的数值实验表明了这一点。在爱森斯坦级数的情况下，我们证明了我们猜想的一个特例。