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arXiv · 2302.04714

Period relations between the Betti-Whittaker periods for ${\rm GL}_n$ under duality

Shih-Yu Chen
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Abstract

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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Shih-Yu Chen. 2024-05-28. Period relations between the Betti-Whittaker periods for ${\rm GL}_n$ under duality. https://doi.org/10.1093/imrn%2Frnae103

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