scholarly journals Stable Conjugacy: Definitions and Lemmas

1979 ◽  
Vol 31 (4) ◽  
pp. 700-725 ◽  
Author(s):  
R. P. Langlands
Keyword(s):  
Trace Formula ◽  
Characteristic Zero ◽  
Present Note ◽  
Zeta Functions ◽  
Base Change ◽  
Shimura Varieties ◽  
Reductive Groups ◽  
Typical Problem

The purpose of the present note is to introduce some notions useful for applications of the trace formula to the study of the principle of functoriality, including base change, and to the study of zeta-functions of Shimura varieties. In order to avoid disconcerting technical digressions I shall work with reductive groups over fields of characteristic zero, but the second assumption is only a matter of convenience, for the problems caused by inseparability are not serious.The difficulties with which the trace formula confronts us are manifold. Most of them arise from the non-compactness of the quotient and will not concern us here. Others are primarily arithmetic and occur even when the quotient is compact. To see how they arise, we consider a typical problem.

Shimura Varieties and the Selberg Trace Formula

1977 ◽  
Vol 29 (6) ◽  
pp. 1292-1299 ◽  
Author(s):  
R. P. Langlands
Keyword(s):  
Trace Formula ◽  
Automorphic Forms ◽  
Zeta Functions ◽  
Higher Dimensions ◽  
Shimura Varieties ◽  
Selberg Trace Formula ◽  
Work In Progress ◽  
Euler Products ◽  
Dimension One

This paper is a report on work in progress rather than a description of theorems which have attained their final form. The results I shall describe are part of an attempt to continue to higher dimensions the study of the relation between the Hasse-Weil zeta-functions of Shimura varieties and the Euler products associated to automorphic forms, which was initiated by Eichler, and extensively developed by Shimura for the varieties of dimension one bearing his name. The method used has its origins in an idea of Sato, which was exploited by Ihara for the Shimura varieties associated to GL(2).

Characters of Non-Connected, Reductive p-Abic Groups

1987 ◽  
Vol 39 (1) ◽  
pp. 149-167 ◽  
Author(s):  
Laurent Clozel
Keyword(s):  
Irreducible Representation ◽  
Characteristic Zero ◽  
Reductive Group ◽  
Base Change ◽  
Reductive Groups ◽  
Local Integrability ◽  
Precise Control ◽  
Distribution Character ◽  
Connected Case

In this paper, we extend to non-connected, reductive groups over p-adic field of characteristic zero Harish-Chandra's theorem on the local integrability of characters.Harish-Chandra's theorem states that the distribution character of an admissible, irreducible representation of a (connected) reductive p-adic group is locally integrable. We show that this extends to any reductive group; just as in the connected case, one even gets a very precise control over the singularities of the character along the singular elements.As will be seen, the proof in the non-connected case is an easy extension of Harish-Chandra's. The reader may wonder why we have bothered to write its generalization completely. The reason is that the original article [8] does not contain proofs for the crucial lemmas, and this makes it impossible to explain why the theorem extends. Because this result is needed for work of Arthur and the author on base change, it has been thought necessary to give complete arguments.

On Connected Components of Shimura Varieties

2002 ◽  
Vol 54 (2) ◽  
pp. 352-395 ◽  
Author(s):  
Thomas J. Haines
Keyword(s):  
Trace Formula ◽  
High Power ◽  
Weighting Factor ◽  
Connected Components ◽  
Shimura Varieties ◽  
Orbital Integrals ◽  
Crucial Step ◽  
Hecke Operator ◽  
Twisted Trace Formula ◽  
Twisted Orbital Integrals

AbstractWe study the cohomology of connected components of Shimura varieties coming from the group GSp2g, by an approach modeled on the stabilization of the twisted trace formula, due to Kottwitz and Shelstad. More precisely, for each character ϖ on the group of connected components of we define an operator L(ω) on the cohomology groups with compact supports Hic(, ), and then we prove that the virtual trace of the composition of L(ω) with a Hecke operator f away from p and a sufficiently high power of a geometric Frobenius , can be expressed as a sum of ω-weighted (twisted) orbital integrals (where ω-weighted means that the orbital integrals and twisted orbital integrals occuring here each have a weighting factor coming from the character ϖ). As the crucial step, we define and study a new invariant α1(γ0; γ, δ) which is a refinement of the invariant α(γ0; γ, δ) defined by Kottwitz. This is done by using a theorem of Reimann and Zink.

scholarly journals On L-Functions of Twisted 3-Dimensional Quaternionic Shimura Varieties

2008 ◽  
Vol 190 ◽  
pp. 87-104
Author(s):  
Cristian Virdol
Keyword(s):  
Galois Group ◽  
Complex Plane ◽  
Zeta Functions ◽  
Shimura Varieties ◽  
Absolute Galois Group ◽  
3 Dimensional ◽  
Entire Complex ◽  
The Absolute

In this paper we compute and continue meromorphically to the entire complex plane the zeta functions of twisted quaternionic Shimura varieties of dimension 3. The twist of the quaternionic Shimura varieties is done by a mod ℘ representation of the absolute Galois group.

scholarly journals Local models for Weil-restricted groups

2016 ◽  
Vol 152 (12) ◽  
pp. 2563-2601 ◽  
Author(s):  
Brandon Levin
Keyword(s):  
Hecke Algebra ◽  
Shimura Varieties ◽  
Reductive Groups ◽  
Local Models ◽  
Central Function ◽  
Nearby Cycles ◽  
Weil Restriction ◽  
Trace Of Frobenius ◽  
Invent Math

We extend the group-theoretic construction of local models of Pappas and Zhu [Local models of Shimura varieties and a conjecture of Kottwitz, Invent. Math.194(2013), 147–254] to the case of groups obtained by Weil restriction along a possibly wildly ramified extension. This completes the construction of local models for all reductive groups when$p\geqslant 5$. We show that the local models are normal with special fiber reduced and study the monodromy action on the sheaves of nearby cycles. As a consequence, we prove a conjecture of Kottwitz that the semi-simple trace of Frobenius gives a central function in the parahoric Hecke algebra. We also introduce a notion of splitting model and use this to study the inertial action in the case of an unramified group.

scholarly journals On Some Generalized Rapoport–Zink Spaces

2019 ◽  
Vol 72 (5) ◽  
pp. 1111-1187
Author(s):  
Xu Shen
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
K3 Surfaces ◽  
Connected Components ◽  
Shimura Varieties ◽  
Reductive Groups ◽  
Mixed Characteristic ◽  
Special Fibers ◽  
Type Spaces ◽  
Local Invariants

AbstractWe enlarge the class of Rapoport–Zink spaces of Hodge type by modifying the centers of the associated $p$-adic reductive groups. Such obtained Rapoport–Zink spaces are said to be of abelian type. The class of Rapoport–Zink spaces of abelian type is strictly larger than the class of Rapoport–Zink spaces of Hodge type, but the two type spaces are closely related as having isomorphic connected components. The rigid analytic generic fibers of Rapoport–Zink spaces of abelian type can be viewed as moduli spaces of local $G$-shtukas in mixed characteristic in the sense of Scholze.We prove that Shimura varieties of abelian type can be uniformized by the associated Rapoport–Zink spaces of abelian type. We construct and study the Ekedahl–Oort stratifications for the special fibers of Rapoport–Zink spaces of abelian type. As an application, we deduce a Rapoport–Zink type uniformization for the supersingular locus of the moduli space of polarized K3 surfaces in mixed characteristic. Moreover, we show that the Artin invariants of supersingular K3 surfaces are related to some purely local invariants.

scholarly journals Functoriality of motivic lifts of the canonical construction

2019 ◽  
Vol 163 (1-2) ◽  
pp. 27-56 ◽  
Author(s):  
Alex Torzewski
Keyword(s):  
Full Subcategory ◽  
Hodge Structure ◽  
Compact Subgroup ◽  
Base Change ◽  
Shimura Varieties ◽  
Open Compact Subgroup ◽  
Chow Motives ◽  
Variation Of Hodge Structure

Abstract Let $$(G,{\mathfrak {X}})$$ ( G , X ) be a Shimura datum and K a neat open compact subgroup of $$G(\mathbb {A}_f)$$ G ( A f ) . Under mild hypothesis on $$(G,{\mathfrak {X}})$$ ( G , X ) , the canonical construction associates a variation of Hodge structure on $$\text {Sh}_K(G,{\mathfrak {X}})(\mathbb {C})$$ Sh K ( G , X ) ( C ) to a representation of G. It is conjectured that this should be of motivic origin. Specifically, there should be a lift of the canonical construction which takes values in relative Chow motives over $$\text {Sh}_K(G,{\mathfrak {X}})$$ Sh K ( G , X ) and is functorial in $$(G,{\mathfrak {X}})$$ ( G , X ) . Using the formalism of mixed Shimura varieties, we show that such a motivic lift exists on the full subcategory of representations of Hodge type $$\{(-1,0),(0,-1)\}$$ { ( - 1 , 0 ) , ( 0 , - 1 ) } . If $$(G,{\mathfrak {X}})$$ ( G , X ) is equipped with a choice of PEL-datum, Ancona has defined a motivic lift for all representations of G. We show that this is independent of the choice of PEL-datum and give criteria for it to be compatible with base change. Additionally, we provide a classification of Shimura data of PEL-type and demonstrate that the canonical construction is applicable in this context.

The Distributions in the Invariant Trace Formula Are Supported on Characters

2000 ◽  
Vol 52 (4) ◽  
pp. 804-814 ◽  
Author(s):  
Robert E. Kottwitz ◽  
Jonathan D. Rogawski
Keyword(s):  
Trace Formula ◽  
Galois Cohomology ◽  
Invariant Form ◽  
Reductive Groups ◽  
Connected Case

AbstractJ. Arthur put the trace formula in invariant form for all connected reductive groups and certain disconnected ones. However his work was written so as to apply to the general disconnected case, modulo two missing ingredients. This paper supplies one of those missing ingredients, namely an argument in Galois cohomology of a kind first used by D. Kazhdan in the connected case.

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