Bessel $F$-isocrystals for reductive groups

Abstract

We construct the Frobenius structure on a rigid connection Be Ǧ on Gm for a split reductive group Ǧ introduced by Frenkel-Gross. These data form a Ǧ-valued overconvergent F -isocrystal Be † Ǧ on G m,Fp , which is the p-adic companion of the Kloosterman Ǧ-local system Kl Ǧ constructed by Heinloth-Ngô-Yun. By studying the structure of the underlying differential equation, we calculate the monodromy group of Be † Ǧ when Ǧ is almost simple (which recovers the calculation of monodromy group of Kl Ǧ due to Katz and Heinloth-Ngô-Yun), and establish functoriality between different Kloosterman Ǧ-local systems as conjectured by Heinloth-Ngô-Yun. We show that the Frobenius Newton polygons of Kl Ǧ are generically ordinary for every Ǧ and are everywhere ordinary on |G m,Fp | when Ǧ is classical or G 2 . Contents 1. Introduction 1.1. Bessel equations and Kloosterman sums 1.2. Generalization for reductive groups 1.3. Strategy of the proof and the organization of the article 2. Review and complements on arithmetic D-modules 2.1. Overconvergent (F -)isocrystals and their rigid cohomologies 2.2. (Co)specialization morphism for de Rham and rigid cohomologies 2.3. Six functors formalism for arithmetic D-modules 2.4. Complements on the cohomology of arithmetic D-modules 2.5. Equivariant holonomic D-modules 2.6. Intermediate extension and the weight theory 2.7. Nearby and vanishing cycles 2.8. Universal local acyclicity 2.9. Local monodromy of an overconvergent F -isocrystal 2.10. Hyperbolic localization for arithmetic D-modules 3. Geometric Satake equivalence for arithmetic D-modules 3.1. The Satake category 3.2. Fusion product 3.3. Hypercohomology functor 3.4. Semi-infinite orbits 3.5. Tannakian structure and the Langlands dual group 3.6. The full Langlands dual group 4. Bessel F -isocrystals for reductive groups 4.1. Kloosterman F -isocrystals for reductive groups 4.2. Comparison between Kl dR Ǧ and Kl rig Ǧ 4.3. Comparison between Kl dR Ǧ and Be Ǧ 4.4. Bessel F -isocrystals for reductive groups 4.5. Monodromy groups

Newton polygons for $L$-functions of generalized Kloosterman sums

“…Actually, for the diagonal Laurent polynomial, Wan [15] got some criterions to decide when the Newton polygon and the Hodge polygon coincide. For the non-diagonal Laurent polynomial, one main technique is Wan's decomposition theorem, that is, decomposing ∆ into small pieces which is diagonal and easy to deal with, the related work can be found in [2], [3], [15], [17], [18]. Further decomposition methods for Newton polygons are developed in [8].…”