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Saturday, April 25, 2020 | History

2 edition of Boundary cohomology of Shimura varieties, III found in the catalog.

Boundary cohomology of Shimura varieties, III

Michael Harris

# Boundary cohomology of Shimura varieties, III

## by Michael Harris

Written in English

Subjects:
• Shimura varieties.,
• Homology theory.

• Edition Notes

The Physical Object ID Numbers Other titles Boundary cohomology of Shimura varieties, three. Statement Michael Harris, Steven Zucker. Series Memoires de la Societe Mathematique de France -- nouvelle serie, no. 85., Mémoire (Société mathématique de France) -- Nouv. sér., no 85. Contributions Zucker, Steven. Pagination 116 p. ; Number of Pages 116 Open Library OL16019843M

ON THE COHOMOLOGY OF SHIMURA VARIETIES 5 as a virtual G(Ap f) × G(Zp) × πf runs through irreducible admissible representations of G(Af), the integer a(πf) is as in [26], p. , and φπp is the local Langlands parameter associated to G(Zp) is a certain maximal compact subgroup of G(Qp). The restriction of the G(Qp)-action to G(Zp) is due to our . This book constructs a candidate for such a local Langlands correspondence on the vanishing cycles attached to the bad reduction over the integer ring of K of a certain family of Shimura varieties. And it proves that this is roughly compatible with the global Galois correspondence realized on the cohomology of the same Shimura varieties. A very recent reference: Umberto Zannier’s book “Some problems of unlikely intersections in arithmetic and geometry”. Andre stated the case for curves in Shimura varieties in , and Oort stated the case of subvarieties´ in A g (moduli space of principally polarised abelian varieties) in The foundations were laid byFile Size: KB.

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### Boundary cohomology of Shimura varieties, III by Michael Harris Download PDF EPUB FB2

Get this from a library. Boundary cohomology of Shimura varieties, III: coherent cohomology on higher-rank boundary strata and applications to Hodge theory. [Michael Harris; Steven W Zucker]. In number theory, a Shimura variety Boundary cohomology of Shimura varieties a higher-dimensional analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over a varieties are not algebraic varieties but are families of algebraic varieties.

Shimura curves are the one-dimensional Shimura varieties. Boundary cohomology of Shimura varieties, II. Hodge theory at the boundary On the boundary cohomology of locally symmetric varieties.

In: Algebraic Geometry Symposium (Saitama Univ.), – () Download references. Authors. Harris. III book author publications. You can also search for this author by:   This book studies the intersection cohomology of the Shimura varieties associated to unitary groups of any rank over Q.

In general, these varieties are not compact. The intersection cohomology of the Shimura variety associated to a reductive group G carries commuting actions of the absolute Galois group of the reflex field and of the group G(Af Author: Sophie Morel.

This book constructs a candidate for such a local Langlands correspondence on the vanishing cycles attached to the bad reduction over the integer ring of K of a certain family of Shimura varieties. And it proves that this is roughly compatible with the global Galois correspondence realized on the cohomology of the same Shimura by: place on the intersection cohomology of the Baily-Borel compacti cation of cer-tain Shimura varieties, and then to stabilize the result for the Shimura varieties associated to unitary groups over Q.

The main result is theorem It expresses the above trace in terms of the twisted trace formula on products of general linear groups, for well.

Etale cohomology 66 5. The cohomology of Igusa varieties 70 Setup 70 A stable trace formula 71 Base change and the twisted trace formula 75 The transfer at p 77 Generic principal series 79 Simple Shimura varieties 85 6. Torsion in the cohomology of unitary Shimura varieties 87 Perverse sheaves on the ag.

The Geometry and Cohomology of Some Simple Shimura III book. (AM) - Ebook written by Michael Harris, Richard Taylor. Read this book using Google Play Books app on your PC, android, iOS devices.

Request PDF | Cohomology of the boundary of Siegel modular varieties of degree two, with applications | Let A2(n) = Γ2(n)\G fraktur sign2 be the quotient of Siegel's space of degree 2 by the.

Let (P,X) be Shimura data, M=M(P,X,K) the Shimura variety of level K. To an algebraic representation of P, one can associate a mixed sheaf (variation of Hodge structure, l-adic sheaf) on M.

Publications and preprints (list) Regulators and characteristic classes (with J. Dupont & R. Hain), The Arithmetic and Geometry of Algebraic Cycles, CRM Proceedings & Lecture No ()-- pdf file Boundary cohomology of Shimura varieties, III: Coherent cohomology on higher-rank boundary strata and applications to Hodge theory (with M.

Harris), Mem. Soc. Math. France 85 ()-- pdf. This book aims first to prove the local Langlands conjecture for GLn over a p-adic field and, second, to identify the action of the decomposition group at a prime of bad reduction on the l-adic cohomology of the "simple" Shimura varieties.

These two problems go hand in hand. The results represent a major advance in algebraic number theory, finally proving the conjecture first proposed in. Book Description: This book aims first to prove the local Langlands conjecture for GL n over a p-adic field and, second, to identify the action of the decomposition group at a prime of bad reduction on the l-adic cohomology of the "simple" Shimura Boundary cohomology of Shimura varieties.

These two problems go hand in hand. The results represent a major advance in algebraic number theory, finally proving the conjecture. on the cohomology of some simple shimura varieties with bad reduction - volume 18 issue 1 - xu shen Skip to main content Accessibility help We use cookies to distinguish you from other users and to provide you with a better experience on our by: 2.

III Some simple Shimura varieties 89 Characteristic zero theory 89 Cohomology 94 The trace formula Integral models IV Igusa varieties IV.

1 Igusa varieties of the first kind IV.2 Igusa varieties of the second kind V Counting Points V.I An application of Fujiwara's trace formula V.2 Honda-Tate File Size: 41KB.

Torsion in the Coherent Cohomology of Shimura Varieties and Galois Representations Abstract We introduce a method for producing congruences between Hecke eigenclasses, possibly torsion, in the coherent cohomology of automorphic vector bundles on certain good reduction Shimura varieties.

tute Summer School entitled “Harmonic Analysis, the Trace Formula, and Shimura Varieties.” It was held at the Fields Institute in Toronto, Canada, from June 2 to J The main goal of the School was to introduce graduate students and young mathematicians to three broad and interrelated areas in the theory of automorphic forms.

[42] Harris, M., Zucker, S.: Boundary cohomology of Shimura varieties, III: Coherent cohomology on higher-rank boundary strata and applications to Hodge theory, Mémoires de la SMF, 85 (). [43] Harris, M., Taylor, R.: The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, ().

Both methods first realize the Hecke eigenvalues of interest in the cohomology with compact support of the open Shimura variety by an analysis of the boundary and then show that they also occur in some space of p-adic cusp forms.

We work with the ordinary locus of the Shimura variety, which for the minimal compactification is by: THE GENERIC PART OF THE COHOMOLOGY OF SHIMURA VARIETIES 3 map again, but this time in a p-adic context with p6= ℓ(whereas [Sch15b] worked in the situation p= ℓ).

We note that this should make it possible to understand the behaviour of ρm at places above ℓ. Remark It is a formal consequence that the Zℓ-cohomology localized at m is. Conférences Paris Pékin Tokyo, Peter Scholze.

Shimura varieties with infinite level, and torsion in the cohomology of locally symmetric spaces. We will discuss the p-adic geometry of Shimura. B, and also results about ramiﬁed places, by using the cohomology of Igusa varieties attached to compact unitary Shimura varieties (cf [Shi1], [Shi2], [Shi3]).

This builds on previous work of Harris and Taylor ([HT]). We give a quick description of the diﬀerent chapters. Chapter 1 contains “known facts” about the ﬁxed point formula.

When. Sir Andrew Wiles - The Abel Lecture - Fermat's Last theorem: abelian and non-abelian approaches - Duration: The Abel Prize 5, views. Author's comments: Although this paper has some merit, it also leaves a lot to be desired, and suffers from the author's inexperience with the material.

For a better treatment of more general questions and corrections to the assertions of this paper, the reader is advised to consult a paper of Thomas Zink, Über die schlechte Reduktion einiger Shimuramannigfaltigkeiten, Comp.

Math. 45 (). $\begingroup$ I would also look at the series of papers by M. Harris and S. Zucker - boundary cohomology of Shimura varieties. In the beginning of part III, they stay that it is standard that the cohomology decomposes as written by Dan, but I do not know where this is proved.

$\endgroup$ – ACL Jan 19 '14 at The workshop will have several goals, including: describing the latest results on the cohomology of Shimura varieties;explaining the key ideas and techniques which underly these results; analyzing the applications of these results to the construction of Galois representations attached to automorphic forms, with the goal of describing the state of the art in this problem.

Following an old suggestion of Clozel, recently realized by Harris-Lan-Taylor-Thorne for characteristic $0$ cohomology classes, one realizes the cohomology of the locally symmetric spaces for $\mathrm{GL}_n$ as a boundary contribution of the cohomology of symplectic or unitary Shimura varieties, so that the key problem is to understand torsion Cited by: Shimura had two children, Tomoko and Haru, with his wife Chikako.

Shimura died on 3 May in Princeton, New Jersey at the age of Research. Shimura was a colleague and a friend of Yutaka Taniyama, with whom he wrote the first book on the complex multiplication of abelian varieties and formulated the Taniyama–Shimura : Guggenheim Fellowship (), Cole.

Boundary Cohomology of Shimura Varieties, III: Coherent Cohomology on Higher-Rank Boundary Strata and Applications to Hodge Theory Michael Harris, Université Paris, Franceand Steven Zucker, Johns Hopkins University, Baltimore, Maryland A publication of the Société Mathématique de France.

In this book, the authors complete the veriﬁcation. Shimura varieties (see ). This phenomenon can be viewed as a modular interpretation for these mixed Shimura varieties. A mixed Shimura variety is pure when the associated algebraic group is reductive.

The treatment of chapter 1 follows [D2] x1. The next problem we have to deal with is how to make mixed Shimura varieties al-gebraic. AN EXAMPLE-BASED INTRODUCTION TO SHIMURA VARIETIES 3 References 65 Index 72 1.

Introduction Shimura varieties are generalizations of modular curves, which have played an important role in many recent developments of number theory. Just to mention a few examples with which this author is more familiar, Shimura varieties wereFile Size: 1MB. We introduce a method for producing congruences between Hecke eigenclasses, possibly torsion, in the coherent cohomology of automorphic vector bundles on certain good reduction Shimura varieties.

The congruences are produced using some generalized Hasse invariants'' adapted to the Ekedahl-Oort stratification of the special fiber. PERFECTOID SHIMURA VARIETIES AND THEIR TORSION COHOMOLOGY, D’APRES SCHOLZE ORG.: M.-H.

NICOLE 1. Introduction The goal of this “groupe de travail” (study group) is to understand Scholze’s preprint [1]. One of its main theorems is to show the existence of torsion Galois representations in the cohomology of Shimura varieties of Hodge type.

Torsion classes in the cohomology of KHT Shimura’s varieties 5 F v the completion of Fat v, O v the ring of integers of F v, $v a uniformizer, q v the cardinal of the residual eld (v) = O v=($ v). Let Bbe a division algebra with center F, of dimension d2 such that at every place xof F.

These are expanded notes prepared for a talk in a learning seminar on Caraiani-Scholze's paper On the generic part of the cohomology of compact unitary Shimura varieties, Spring at summarize the major ingredients of the proof, explain the preservation of perversity under the Hodge-Tate period map and deduce the main theorems: 1) the existence of Galois representations.

E.g. already in the case of Hilbert modular varieties (or their compact cousins, coming from quat. algebras, if you prefer), in some situations one expects a reducible Galois rep'n, and these are not known to be semi-simple, I believe.

Speaker: George Boxer, University of Chicago Title: Coherent cohomology of Shimura varieties and derived Hecke algebras Abstract: James Newton and Jack Thorne have defined "derived Hecke algebras" which act on the cohomology of arithmetic locally symmetric spaces at the derived category use the work of Scholze to show that in suitable situations there are Galois representations.

M.H.'s research received funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/) / ERC Grant agreement number (AAMOT). K.-W.L.'s research was partially supported by NSF Grants DMS, DMS, and DMS, and by an Alfred P. Sloan Research Fellowship.

R.T.'s research was partially supported by Cited by: MATTHEW GREENBERG deﬁned as follows. For D ∈ Div0 H p,letg D denote a rational function on P1(C p) with divisor E be the Tate period of E, and set D 1 ⊗D 2,ϕ= P1(Q p) log q E g D 2 (t)dϕ(D1)(t).

Assume the following Heegner hypothesis: All primes divisors of N/pare split in τ ∈ K ∩H the stabilizer τ is an abelian group of rank one; let γ τ be a generator. MIXED STRUCTURES IN SHIMURA VARIETIES AND AUTOMORPHIC FORMS ARVIND N.

NAIR Abstract. We study the mixed structures appearing in the cohomology of noncompact Shimura varieties using a new spectral sequence which arises naturally from Morel’s theory of weight truncations [M08].

The E 1 term of this spectral sequence is a direct sum of pure. Introduction to Shimura Varieties 1. Introduction Shimura varieties play a crucial role for providing a link between two major aspects of modern number theory: the automorphic and arithmetic aspect.

The interplay between these two dates back to the midth century with the work of Kronecker and Kummer. It has later been taken.Cohomology of line bundles on abelian varieties 12 An application of G.I.T 13 Spreading the abelian scheme structure 16 Smoothness 17 Adelic description and Hecke correspondences 18 3.

Shimura varieties of PEL type 20 Shimura varieties. File Size: KB.the cohomology of Shimura varieties, speci cally at places of bad reduction. Let us rst brie y recall the expected description, due to Langlands.

Let Sh K be some Shimura variety associated to a reductive group G over Q and a compact open subgroup KˆG(A f), and some additional data. It is de ned over a number eld E, canonically embedded into C Author: Peter Scholze, Sug Woo Shin.