Automorphic functions as the trace of Frobenius

Cornell University - arXiv, 2020

We identify the category Shv Nilp (BunG) of automorphic sheaves with nilpotent singular support with its own dual, and relate this structure to the Serre functor on Shv Nilp (BunG) and miraculous duality. D. ARINKIN, D. GAITSGORY, D. KAZHDAN, S. RASKIN, N. ROZENBLYUM, Y. VARSHAVSKY 4.2. Kernels defined by objects from Shv Nilp (BunG) 4.3. Pairings against sheaves with nilpotent singular support, revisited 4.4. A refined version of Theorem 1.3.7 4.5. A converse statement 4.6. A version for a universally Nilp-cotruncative quasi-compact open substack 4.7. Proof of Theorem 4.5.2 5. Serre functor on Shv Nilp (BunG) 5.1. The Serre functor 5.2. Serre vs pseudo-identity 5.3. Duality-adapted pairs, complements 5.4. Constraccessible pairs, complements 5.5. Serre pairs 5.6. Relation to the miraculous functor 5.7. Relation to the miraculous functor, continued 5.8. The (non)-Serre property of BunG Appendix A. Sheaves on stacks A.1. The basics A.2. Verdier-compatible stacks A.3. Base change maps A.4. Verdier self-duality A.5. Specifying singular support A.6. Categorical Künneth formulas Appendix B. Functors defined by kernels and miraculous duality B.1. Functors defined by kernels B.2. A discontinuous version B.3. Functors co-defined by kernels B.4. The miraculous functor B.5. Adjunctions for functors defined by kernels B.6. Adjunctions and Verdier duality B.7. Miraculous stacks B.8. A criterion for admitting an adjoint B.9. Proof of Theorem B.8.8 B.10. An example: the ULA property Appendix C. Sheaves on non quasi-compact algebraic stacks C.1. Cotruncative substacks C.2. The non-quasi-compact case: the "co"-category C.3. Verdier duality in the non-quasi-compact case C.4. Functors defined by kernels in the non-quasi-compact case C.5. Miraculous stacks-the non quasi compact case References

2020

The goal of this note is to supply proofs of two theorems stated [AGKRRV], which deal with the Local Term map for the Frobenius endomorphism of Artin stacks of finite type over Fq. Namely, we show that the “true local terms” of the Frobenius endomorphism coincide with the “naive local terms” and that the “naive local terms” commute with !-pushforwards. The latter result is a categorical version of the classical Grothendieck–Lefschetz trace formula.

Advances in Mathematics, 1976

arXiv (Cornell University), 2005

In this paper we provide a geometric framework for the study of characters of depth-zero representations of unramified groups over local fields with finite residue fields which is built directly on Lusztig's theory of character sheaves for groups over finite fields and uses ideas due to Schneider-Stuhler. Specifically, we introduce a class of coefficient systems on Bruhat-Tits buildings of perverse sheaves sheaves on affine algebraic groups over an algebraic closure of a finite field, and to each supercuspidal depth-zero representation of an unramified p-adic group we associate a formal sum of these coefficient systems, called a model for the representation. Then, using a character formula due to Schneider-Stuhler and a fixed-point formula in etale cohomology we show that each model defines a distribution which coincides with the Harish-Chandra character of the corresponding representation, on the set of regular elliptic elements. The paper includes a detailed treatment of SL(2), Sp(4) and GL(n) as examples of the theory. 1991 Mathematics Subject Classification. 22E50 (Representations of Lie and linear algebraic groups over local fields); 20G40 (Linear algebraic groups over finite fields). Key words and phrases. p-adic groups; depth-zero representations; character sheaves. * * * A.-M.A. wishes to thank the University of Calgary for providing the opportunity for the authors to work together on this project. She also wishes to thank the Pacific Institute of Mathematical Science and the Institute for Advanced Study (Princeton) for their support. This work was presented by A.-M.A. at Ann Arbor at the invitation of Julee Kim and Gopal Prasad and in Berlin (von Humboldt University) at the invitation of Volker Heiermann and Wilhem Zink, and also at the meeting on K-types for p-adic groups at Münster organized by Guy Henniart, Philip Kutzko and Peter Schneider. She thanks Roger Carter, Ivan Fesenko and Michel Gros for their interest in this work. C.C. wishes to thank theÉcole Normale Supérieure, the Centre National de la Recherche Scientifique, the Institut des HautesÉtudes Scientifiques and the National Science and Engineering Research Council (Canada) for support during various parts of the writing of this paper. He also wishes to thank Jonathan Korman for help and Sam Evans for his encouragement and hospitality. This paper was presented in abbreviated form by C.C. in the Summer of 2004 at Harvard at the invitation of Laurent Berger and then at a Fields Institute Workshop at the University of Ottawa at the invitation of Monica Nevins. Finally, both authors thank Peter Schneider for helpful conversations. Contents 1.10. Categories 17 2. Cuspidal coefficient systems 18 2.1. Parabolic restriction 18 2.2. Weakly-equivariant objects 21 2.3. Cuspidal coefficient systems 25 3. Admissible coefficient systems 32 3.1. Parabolic subgroups 32 3.2. Local parabolic induction 34 3.3. Parabolic induction 35 3.4. Transitive parabolic induction 39 3.5. Admissible coefficient systems 40 4. Enters Frobenius 41 4.1. Frobenius-stable coefficient systems 41 4.2. Parabolic restriction and frobenius 44 4.3. Cuspidal coefficient systems revisited 45 4.4. Parabolic induction and frobenius 45 5. Supercuspidal depth-zero representations 47 5.1. Characteristic functions 47 5.2. Models for representations 49 5.3. Distributions associated to admissible coefficient systems 52 6. Examples 52 6.1. SL(2) 52 6.2. Sp(4) 56 6.3. GL(n) 63 References 65 Recall the derived category D b c (Ḡ i ,Q ℓ) of cohomologically bounded constructible ℓ-adic sheaves with ℓ = p, introduced in [Del80, 1.1.

arXiv (Cornell University), 2007

Given a local ring of positive prime characteristic there is a natural Frobenius action on its local cohomology modules with support at its maximal ideal. In this paper we study the local rings for which the local cohomology modules have only finitely many submodules invariant under the Frobenius action. In particular we prove that F-pure Gorenstein local rings as well as the face ring of a finite simplicial complex localized or completed at its homogeneous maximal ideal have this property. We also introduce the notion of an antinilpotent Frobenius action on an Artinian module over a local ring and use it to study those rings for which the lattice of submodules of the local cohomology that are invariant under Frobenius satisfies the Ascending Chain Condition.