对偶下 ${\rm GL}_n$ 的 Betti-Whittaker 周期之间的周期关系
Period relations between the Betti-Whittaker periods for ${\rm GL}_n$ under duality
AI总结：
本文在正则性条件下证明了对偶下 ${\rm GL}_n$ 的 Betti-Whittaker 周期关系，并由此得到正交型相对周期的平凡性，进而推出临界 $L$-值之比的代数性。
摘要：
本文在若干正则性条件下，证明了与 ${\rm GL}_n(\mathbb{A})$ 的正则代数尖点自守表示及其逆步表示相关联的 Betti-Whittaker 周期之间的周期关系。作为推论，我们得到了与 ${\rm GL}_{2n}(\mathbb{A})$ 的正交型正则代数尖点自守表示相关联的相对周期的平凡性，这结合 Harder 和 Raghuram 的结果，蕴含了 ${\rm GSpin}_{2n}^* \times {\rm GL}_{n'}$ 的相继临界 $L$-值之比的代数性。
英文摘要：
In this paper, under some regularity conditions, we prove a period relation between the Betti--Whittaker periods associated to a regular algebraic cuspidal automorphic representation of ${\rm GL}_n(\mathbb{A})$ and its contragredient. As a consequence, we obtain the trivialness of the relative period associated to a regular algebraic cuspidal automorphic representation of ${\rm GL}_{2n}(\mathbb{A})$ of orthogonal type, which implies the algebraicity of the ratios of successive critical $L$-values for ${\rm GSpin}_{2n}^* \times {\rm GL}_{n'}$ by the result of Harder and Raghuram.
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