scholarly journals Cohomologie d’intersection des variétés modulaires de Siegel, suite

2011 ◽  
Vol 147 (6) ◽  
pp. 1671-1740 ◽  
Author(s):  
Sophie Morel
Keyword(s):  
Fixed Point ◽  
Trace Formula ◽  
Compact Support ◽  
Automorphic Forms ◽  
Shimura Varieties ◽  
Academic Press ◽  
Test Functions ◽  
Hecke Operator ◽  
Frobenius Endomorphism ◽  
Modular Varieties

AbstractIn this work, we study the intersection cohomology of Siegel modular varieties. The goal is to express the trace of a Hecke operator composed with a power of the Frobenius endomorphism (at a good place) on this cohomology in terms of the geometric side of Arthur’s invariant trace formula for well-chosen test functions. Our main tools are the results of Kottwitz about the contribution of the cohomology with compact support and about the stabilization of the trace formula, Arthur’s L2 trace formula and the fixed point formula of Morel [Complexes pondérés sur les compactifications de Baily–Borel. Le cas des variétés de Siegel, J. Amer. Math. Soc. 21 (2008), 23–61]. We ‘stabilize’ this last formula, i.e. express it as a sum of stable distributions on the general symplectic groups and its endoscopic groups, and obtain the formula conjectured by Kottwitz in [Shimura varieties and λ-adic representations, in Automorphic forms, Shimura varieties and L-functions, Part I, Perspectives in Mathematics, vol. 10 (Academic Press, San Diego, CA, 1990), 161–209]. Applications of the results of this article have already been given by Kottwitz, assuming Arthur’s conjectures. Here, we give weaker unconditional applications in the cases of the groups GSp4 and GSp6.

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.

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).

scholarly journals The spectral side of Arthur's trace formula

2009 ◽  
Vol 106 (37) ◽  
pp. 15563-15566 ◽  
Author(s):  
Tobias Finis ◽  
Erez M. Lapid ◽  
Werner Müller
Keyword(s):  
Explicit Expression ◽  
Number Field ◽  
Trace Formula ◽  
Wide Class ◽  
Automorphic Forms ◽  
Absolute Convergence ◽  
Reductive Group ◽  
Spectral Expansion ◽  
Test Functions ◽  
Hyperbolic Surfaces

The trace formula is one of the most important tools in the theory of automorphic forms. It was invented in the 1950's by Selberg, who mostly studied the case of hyperbolic surfaces, and was later on developed extensively by Arthur in the generality of an adelic quotient of a reductive group over a number field. Here we provide an explicit expression for the spectral side, improving Arthur's fine spectral expansion. As a result, we obtain its absolute convergence for a wide class of test functions.

AUTOMORPHIC LEFSCHETZ PROPERTIES FOR NONCOMPACT ARITHMETIC MANIFOLDS

2021 ◽  
pp. 1-48
Author(s):  
Arvind N. Nair ◽  
Ankit Rai
Keyword(s):  
Symmetric Spaces ◽  
Automorphic Forms ◽  
Hodge Theory ◽  
Hyperbolic Manifolds ◽  
Shimura Varieties ◽  
Restriction Maps ◽  
Locally Symmetric Spaces ◽  
Compact Congruence ◽  
Lefschetz Properties ◽  
Invent Math

Abstract We prove the injectivity of Oda-type restriction maps for the cohomology of noncompact congruence quotients of symmetric spaces. This includes results for restriction between (1) congruence real hyperbolic manifolds, (2) congruence complex hyperbolic manifolds, and (3) orthogonal Shimura varieties. These results generalize results for compact congruence quotients by Bergeron and Clozel [Quelques conséquences des travaux d’Arthur pour le spectre et la topologie des variétés hyperboliques, Invent. Math.192 (2013), 505–532] and Venkataramana [Cohomology of compact locally symmetric spaces, Compos. Math.125 (2001), 221–253]. The proofs combine techniques of mixed Hodge theory and methods involving automorphic forms.

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.

scholarly journals The Plectic Weight Filtration on Cohomology of Shimura Varieties and Partial Frobenius

2021 ◽  
Vol 9 ◽  
Author(s):  
Zhiyou Wu
Keyword(s):  
Shimura Varieties ◽  
Weight Filtration ◽  
Hilbert Modular ◽  
Hilbert Modular Varieties ◽  
Modular Varieties

Abstract We prove that there is a natural plectic weight filtration on the cohomology of Hilbert modular varieties in the spirit of Nekovář and Scholl. This is achieved with the help of Morel’s work on weight t-structures and a detailed study of partial Frobenius. We prove in particular that the partial Frobenius extends to toroidal and minimal compactifications.

scholarly journals Level-raising for Saito–Kurokawa forms

2009 ◽  
Vol 145 (4) ◽  
pp. 915-953
Author(s):  
Claus M. Sorensen
Keyword(s):  
Trace Formula ◽  
Modular Forms ◽  
Automorphic Forms ◽  
Automorphic Representation ◽  
Weil Representation ◽  
Local Component ◽  
Fundamental Lemma ◽  
Error Terms ◽  
Stable Trace Formula ◽  
Steinberg Representation

AbstractThis paper provides congruences between unstable and stable automorphic forms for the symplectic similitude group GSp(4). More precisely, we raise the level of certain CAP representations Π arising from classical modular forms. We first transfer Π to π on a suitable inner form G; this is achieved by θ-lifting. For π, we prove a precise level-raising result that is inspired by the work of Bellaiche and Clozel and which relies on computations of Schmidt. We thus obtain a $\tilde {\pi }$ congruent to π, with a local component that is irreducibly induced from an unramified twist of the Steinberg representation of the Klingen parabolic. To transfer $\tilde {\pi }$ back to GSp(4), we use Arthur’s stable trace formula. Since $\tilde {\pi }$ has a local component of the above type, all endoscopic error terms vanish. Indeed, by results due to Weissauer, we only need to show that such a component does not participate in the θ-correspondence with any GO(4); this is an exercise in using Kudla’s filtration of the Jacquet modules of the Weil representation. We therefore obtain a cuspidal automorphic representation $\tilde {\Pi }$ of GSp(4), congruent to Π, which is neither CAP nor endoscopic. It is crucial for our application that we can arrange for $\tilde {\Pi }$ to have vectors fixed by the non-special maximal compact subgroups at all primes dividing N. Since G is necessarily ramified at some prime r, we have to show a non-special analogue of the fundamental lemma at r. Finally, we give an application of our main result to the Bloch–Kato conjecture, assuming a conjecture of Skinner and Urban on the rank of the monodromy operators at the primes dividing N.

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