Bessel F-isocrystals for reductive groups

Daxin Xu Morningside Center of Mathematics and Hua Loo-Keng Key Laboratory of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China Xinwen Zhu Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA

Number Theory Algebraic Geometry mathscidoc:2203.24005

Inventiones Mathematicae, 227, 997-1092, 2022.1
We construct the Frobenius structure on a rigid connection Be_{\hat G} on G_m for a split reductive group \hat G introduced by Frenkel-Gross. These data form a \hat G-valued overconvergent F-isocrystal Be^†_{\hat G} on G_{m,F_p}, which is the p-adic companion of the Kloosterman \hat G-local system Kl_{\hat G} constructed by Heinloth-Ngô-Yun. By studying the structure of the underlying differential equation, we calculate the monodromy group of Be^†_{\hat G} when \hat G is almost simple (which recovers the calculation of monodromy group of Kl_{\hat G} due to Katz and Heinloth–Ngô–Yun), and prove a conjecture of Heinloth-Ngô-Yun on the functoriality between different Kloosterman \hat G-local systems. We show that the Frobenius Newton polygons of Kl_{\hat G} are generically ordinary for every \hat G and are everywhere ordinary on |G_{m,F_p}| when \hat G is classical or G_2.
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  title={Bessel F-isocrystals for reductive groups},
  author={Daxin Xu, and Xinwen Zhu},
  booktitle={Inventiones Mathematicae},
Daxin Xu, and Xinwen Zhu. Bessel F-isocrystals for reductive groups. 2022. Vol. 227. In Inventiones Mathematicae. pp.997-1092.
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