Mathematische Zeitschrift

, Volume 279, Issue 1–2, pp 61–84 | Cite as

On linear periods

Article
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Abstract

Let \(\pi '\) be a cuspidal automorphic representation of \({\mathrm{GL}}_{2n}({\mathbb {A}})\), which is assumed to be the Jacquet–Langlands transfer from a cuspidal automorphic representation \(\pi \) of \({\mathrm{GL}}_{2m}(D)({\mathbb {A}})\), where \(D\) is a division algebra so that \({\mathrm{GL}}_{2m}(D)\) is an inner form of \({\mathrm{GL}}_{2n}\). In this paper, we consider the relation between linear periods on \(\pi \) and \(\pi '\). We conjecture that the non-vanishing of the linear period on \(\pi \) would imply the non-vanishing of that on \(\pi '\). We illustrate an approach using a relative trace formula towards this conjecture, and prove the existence of smooth transfer over non-archimedean local fields.

Mathematics Subject Classification

11F70 11F72 
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Notes

Acknowledgments

This work was supported by the National Key Basic Research Program of China (No. 2013CB834202). The author would like to thank Dipendra Prasad for his valuable comments, and Wen-Wei Li for his long list of useful comments and suggestions. He also thanks Dihua Jiang and Binyong Sun for helpful discussions. He expresses gratitude to Ye Tian and Linsheng Yin for their constant encouragement and support. The anonymous referee pointed out a gap and numerous mathematical and grammatical inaccuracies, made a lot of useful comments, and helped the author to greatly improve the exposition. The author is grateful to him or her.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.School of Mathematical SciencesBeijing Normal UniversityBeijingPeople’s Republic of China

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