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Hodge numbers of motives attached to Kloosterman and Airy moments

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Published/Copyright: January 30, 2024

Abstract

Fresán, Sabbah, and Yu constructed motives M n + 1 k ( Kl ) over encoding symmetric power moments of Kloosterman sums in n variables. When n = 1 , they use the irregular Hodge filtration on the exponential mixed Hodge structure associated with M 2 k ( Kl ) to compute the Hodge numbers of M 2 k ( Kl ) , which turn out to be either 0 or 1. In this article, I explain how to compute the (irregular) Hodge numbers of M n + 1 k ( Kl ) for n = 2 or for general values of n such that gcd ( k , n + 1 ) = 1 . I will also discuss related motives attached to Airy moments constructed by Sabbah and Yu. In particular, the computation shows that there are Hodge numbers bigger than 1 in most cases.

Acknowledgements

This work is based on the author’s Ph.D. thesis completed at Centre de Mathématiques Laurent Schwartz in École Polytechnique. The author thanks his supervisors, Javier Fresán and Claude Sabbah, for proposing this question to him and for their guidance and fruitful discussions. The author also thanks Alberto Castaño Domínguez, Lei Fu, and Christian Sevenheck for their feedback on a previous version of this article and to Gabriel Ribeiro, Bin Wang, Jeng-Daw Yu, and Bingyu Zhang for valuable discussions. Lastly, the author appreciates an anonymous referee for numerous suggestions to correct inaccuracies and enhance this paper’s presentation.

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Received: 2023-02-16
Revised: 2023-10-30
Published Online: 2024-01-30
Published in Print: 2024-03-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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