A stable trace formula for Igusa varieties

2010 ◽  
Vol 9 (4) ◽  
pp. 847-895 ◽  
Author(s):  
Sug Woo Shin
Keyword(s):  
Trace Formula ◽  
Positive Characteristic ◽  
Galois Representations ◽  
Type A ◽  
Shimura Varieties ◽  
Explicit Method ◽  
Good Reduction ◽  
Orbital Integrals ◽  
Fundamental Lemma ◽  
Stable Trace Formula

AbstractIgusa varieties are smooth varieties in positive characteristic p which are closely related to Shimura varieties and Rapoport–Zink spaces. One motivation for studying Igusa varieties is to analyse the representations in the cohomology of Shimura varieties which may be ramified at p. The main purpose of this work is to stabilize the trace formula for the cohomology of Igusa varieties arising from a PEL datum of type (A) or (C). Our proof is unconditional thanks to the recent proof of the fundamental lemma by Ngô, Waldspurger and many others.An earlier work of Kottwitz, which inspired our work and proves the stable trace formula for the special fibres of PEL Shimura varieties with good reduction, provides an explicit way to stabilize terms at ∞. Stabilization away from p and ∞ is carried out by the usual Langlands–Shelstad transfer as in work of Kottwitz. The key point of our work is to develop an explicit method to handle the orbital integrals at p. Our approach has the technical advantage that we do not need to deal with twisted orbital integrals or the twisted fundamental lemma.One application of our formula, among others, is the computation of the arithmetic cohomology of some compact PEL-type Shimura varieties of type (A) with non-trivial endoscopy. This is worked out in a preprint of the author's entitled ‘Galois representations arising from some compact Shimura varieties’.

On the Fundamental Lemma for Standard Endoscopy: Reduction to Unit Elements

1995 ◽  
Vol 47 (5) ◽  
pp. 974-994 ◽  
Author(s):  
Thomas C. Hales
Keyword(s):  
Trace Formula ◽  
Simple Form ◽  
Hecke Algebras ◽  
Orbital Integrals ◽  
Fundamental Lemma ◽  
Stable Trace Formula

AbstractThe fundamental lemma for standard endoscopy follows from the matching of unit elements in Hecke algebras. A simple form of the stable trace formula, based on the matching of unit elements, shows the fundamental lemma to be equivalent to a collection of character identities. These character identities are established by comparing them to a compact-character expansion of orbital integrals.

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 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.

scholarly journals Transfert d’intégrales orbitales pour le groupe métaplectique

2010 ◽  
Vol 147 (2) ◽  
pp. 524-590 ◽  
Author(s):  
Wen-Wei Li
Keyword(s):  
Trace Formula ◽  
Local Field ◽  
Transfer Factor ◽  
Characteristic Zero ◽  
Prior Work ◽  
Selberg Trace Formula ◽  
Metaplectic Groups ◽  
Orbital Integrals ◽  
Fundamental Lemma ◽  
Set Up

AbstractWe set up a formalism of endoscopy for metaplectic groups. By defining a suitable transfer factor, we prove an analogue of the Langlands–Shelstad transfer conjecture for orbital integrals over any local field of characteristic zero, as well as the fundamental lemma for units of the Hecke algebra in the unramified case. This generalizes prior work of Adams and Renard in the real case and serves as a first step in studying the Arthur–Selberg trace formula for metaplectic groups.

scholarly journals On a conjecture of Pappas and Rapoport about the standard local model for GL_d

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Dinakar Muthiah ◽  
Alex Weekes ◽  
Oded Yacobi
Keyword(s):  
Type A ◽  
Positive Answer ◽  
Local Model ◽  
Schubert Varieties ◽  
Shimura Varieties ◽  
Local Models ◽  
Full Generality ◽  
Frobenius Splitting ◽  
Affine Grassmannians ◽  
Main Ideas

AbstractIn their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of {n\times n} matrices. We give a positive answer to their conjecture in full generality. Our main ideas follow naturally from two of our previous works. The first is our proof of a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman on the equations defining type A affine Grassmannians. The second is the work of the first two authors and Kamnitzer on affine Grassmannian slices and their reduced scheme structure. We also present a version of our argument that is almost completely elementary: the only non-elementary ingredient is the Frobenius splitting of Schubert varieties.

scholarly journals RAPOPORT–ZINK UNIFORMIZATION OF HODGE-TYPE SHIMURA VARIETIES

2018 ◽  
Vol 6 ◽  
Author(s):  
WANSU KIM
Keyword(s):  
Shimura Varieties ◽  
Good Reduction ◽  
Canonical Models

We show that the integral canonical models of Hodge-type Shimura varieties at odd good reduction primes admits ‘$p$-adic uniformization’ by Rapoport–Zink spaces of Hodge type constructed in Kim [Forum Math. Sigma6(2018) e8, 110 MR 3812116].

scholarly journals Constructing elliptic curves from Galois representations

2018 ◽  
Vol 154 (10) ◽  
pp. 2045-2054
Author(s):  
Andrew Snowden ◽  
Jacob Tsimerman
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Representations ◽  
Good Reduction ◽  
Arithmetic Surface

Given a non-isotrivial elliptic curve over an arithmetic surface, one obtains a lisse $\ell$-adic sheaf of rank two over the surface. This lisse sheaf has a number of straightforward properties: cyclotomic determinant, finite ramification, rational traces of Frobenius elements, and somewhere not potentially good reduction. We prove that any lisse sheaf of rank two possessing these properties comes from an elliptic curve.

scholarly journals ON THE COHOMOLOGY OF SOME SIMPLE SHIMURA VARIETIES WITH BAD REDUCTION

2016 ◽  
Vol 18 (1) ◽  
pp. 1-24
Author(s):  
Xu Shen
Keyword(s):  
Galois Representations ◽  
Shimura Varieties

We determine the Galois representations inside the$\ell$-adic cohomology of some quaternionic and related unitary Shimura varieties at ramified places. The main results generalize the previous works of Reimann and Kottwitz in this setting to arbitrary levels at$p$, and confirm the expected description of the cohomology due to Langlands and Kottwitz.

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