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Algebraicity of the central critical values of twisted triple product L-functions

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Abstract

We study the algebraicity of the central critical values of twisted triple product L-functions associated to motivic Hilbert cusp forms over a totally real étale cubic algebra in the totally unbalanced case. The algebraicity is expressed in terms of the cohomological period constructed via the theory of coherent cohomology on quaternionic Shimura varieties developed by Harris. As an application, we generalize our previous result with Cheng on Deligne’s conjecture for certain automorphic L-functions for \({\text {GL}}_3 \times {\text {GL}}_2\).

Résumé

Nous étudions l’algébraicité des valeurs critiques centrales des fonctions L à triple produit torsadé associées aux formes de cuspides de Hilbert motiviques sur une algèbre cubique étale totalement réelle dans le cas totalement déséquilibré. L’algébraicité est exprimée en termes de période cohomologique construite via la théorie de la cohomologie cohérente sur les variétés quaternioniques de Shimura développée par Harris. Comme application, nous généralisons notre résultat précédent avec Cheng sur la conjecture de Deligne pour certaines fonctions L automorphes pour \({\text {GL}}_3 \times {\text {GL}}_2\).

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Acknowledgements

This work was initiated during the author’s visit to Kyoto University supported by the Graduate Students Study Abroad Program sponsored by Ministry of Science and Technology. The author would like to thank Atsushi Ichino for suggesting the problem and his advice and encouragement. Finally, the author is grateful to the referees for the suggestions and comments on the improvement of the manuscript.

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  1. Institute of Mathematics, Academia Sinica, 6F, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei, 10617, Taiwan, ROC

    Shih-Yu Chen

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Chen, SY. Algebraicity of the central critical values of twisted triple product L-functions. Ann. Math. Québec 47, 403–442 (2023). https://doi.org/10.1007/s40316-021-00169-3

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