AI 中文总结
本文推广了Betti-Whittaker周期在对偶性下的行为结果，应用于三重积L函数和扭曲Asai L函数的特殊值有理性结果，并给出了正交群L函数特殊值的新证明。
AI 中文摘要
一位作者（陈）之前曾证明了一个关于在某些正则性假设下，对于GL_n/Q的上同调 cuspidal 自动表示的Betti-Whittaker周期在对偶性下的行为的结果，而他在证明中使用了这些表示与L值的关系作为锚点。在本文中，我们证明了这一结果的推广，适用于任何数域F上的GL_n，且不假定任何正则性假设，也不依赖L值，而是使用GL_n的外自同构作为主要工具。然后，利用Harder和另一位作者（Raghuram）的结果，我们将该结果应用于一般三重积L函数特殊值比值的新有理性结果，以及一般扭曲Asai L函数的特殊值。我们还给出了Bhagwat和Raghuram之前关于正交群L函数特殊值的结果的新证明。我们还给出了Betti-Shalika周期在对偶性下的周期关系的变体，以及Betti-Whittaker周期在F的Galois自同构下的行为。
英文摘要
One of the authors (Chen) had previously proved a result on the behavior of Betti-Whittaker periods under duality for cohomological cuspidal automorphic representations of ${\rm GL}_n/{\mathbb Q}$ under some regularity assumptions while using their relation to $L$-values as an anchor in his proof. In this article we prove a generalization of this result to ${\rm GL}_n$ over any number field $F$ without any regularity assumptions and without recourse to $L$-values, while using the outer-automorphism of ${\rm GL}_n$ as the main tool. Then, using results of Harder and one of the other authors (Raghuram), we give applications to new rationality results for the ratios of special values of general triple product $L$-functions and for general twisted Asai $L$-functions. We also give a new proof of a previous result of Bhagwat and Raghuram on the special values of $L$-functions for orthogonal groups. We present variations on period relations for the Betti-Shalika periods under duality, and the behavior of Betti-Whittaker periods under Galois automorphisms of $F$.
