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#### Number TheoryAlgebraic Geometrymathscidoc: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.
@inproceedings{daxin2022bessel,
title={Bessel F-isocrystals for reductive groups},
author={Daxin Xu, and Xinwen Zhu},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220317162759367707999},
booktitle={Inventiones Mathematicae},
volume={227},
pages={997-1092},
year={2022},
}

Daxin Xu, and Xinwen Zhu. Bessel F-isocrystals for reductive groups. 2022. Vol. 227. In Inventiones Mathematicae. pp.997-1092. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220317162759367707999.