Drinfeld Moduli Schemes And Automorphic Forms

Author : Yuval Z Flicker
Publisher : Springer Science & Business Media
Page : 150 pages
File Size : 49,6 Mb
Release : 2013-01-04
Category : Mathematics
ISBN : 9781461458883

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Drinfeld Moduli Schemes and Automorphic Forms by Yuval Z Flicker Pdf

Drinfeld Moduli Schemes and Automorphic Forms: The Theory of Elliptic Modules with Applications is based on the author’s original work establishing the correspondence between ell-adic rank r Galois representations and automorphic representations of GL(r) over a function field, in the local case, and, in the global case, under a restriction at a single place. It develops Drinfeld’s theory of elliptic modules, their moduli schemes and covering schemes, the simple trace formula, the fixed point formula, as well as the congruence relations and a "simple" converse theorem, not yet published anywhere. This version, based on a recent course taught by the author at The Ohio State University, is updated with references to research that has extended and developed the original work. The use of the theory of elliptic modules in the present work makes it accessible to graduate students, and it will serve as a valuable resource to facilitate an entrance to this fascinating area of mathematics.

Author : M Van Der Put,E-u Gekeler,M Reversat,Jan Van Geel
Publisher : World Scientific
Page : 378 pages
File Size : 52,9 Mb
Release : 1997-08-27
Category : Mathematics
ISBN : 9789814546409

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Drinfeld Modules, Modular Schemes And Applications by M Van Der Put,E-u Gekeler,M Reversat,Jan Van Geel Pdf

In his 1974 seminal paper 'Elliptic modules', V G Drinfeld introduced objects into the arithmetic geometry of global function fields which are nowadays known as 'Drinfeld Modules'. They have many beautiful analogies with elliptic curves and abelian varieties. They study of their moduli spaces leads amongst others to explicit class field theory, Jacquet-Langlands theory, and a proof of the Shimura-Taniyama-Weil conjecture for global function fields.This book constitutes a carefully written instructional course of 12 lectures on these subjects, including many recent novel insights and examples. The instructional part is complemented by research papers centering around class field theory, modular forms and Heegner points in the theory of global function fields.The book will be indispensable for everyone who wants a clear view of Drinfeld's original work, and wants to be informed about the present state of research in the theory of arithmetic geometry over function fields.

Author : Ernst-Ulrich Gekeler
Publisher : World Scientific Publishing Company Incorporated
Page : 361 pages
File Size : 47,6 Mb
Release : 1997
Category : Mathematics
ISBN : 9810230672

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Proceedings of the Workshop on Drinfeld Modules, Modular Schemes and Applications by Ernst-Ulrich Gekeler Pdf

Author : Thomas Lehmkuhl
Publisher : American Mathematical Soc.
Page : 113 pages
File Size : 46,9 Mb
Release : 2009-01-21
Category : Science
ISBN : 9780821842447

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Compactification of the Drinfeld Modular Surfaces by Thomas Lehmkuhl Pdf

In this article the author describes in detail a compactification of the moduli schemes representing Drinfeld modules of rank 2 endowed with some level structure. The boundary is a union of copies of moduli schemes for Drinfeld modules of rank 1, and its points are interpreted as Tate data. The author also studies infinitesimal deformations of Drinfeld modules with level structure.

Author : Anonim
Publisher : Cambridge University Press
Page : 382 pages
File Size : 54,6 Mb
Release : 2024-07-01
Category : Electronic
ISBN : 9780521470612

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Cohomology of Drinfeld Modular Varieties, Part 2, Automorphic Forms, Trace Formulas and Langlands Correspondence by Anonim Pdf

Author : Ernst-Ulrich Gekeler
Publisher : Springer
Page : 122 pages
File Size : 40,5 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540473862

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Drinfeld Modular Curves by Ernst-Ulrich Gekeler Pdf

Author : Flicker Yuval Z
Publisher : World Scientific
Page : 340 pages
File Size : 40,8 Mb
Release : 2005-08-15
Category : Electronic
ISBN : 9789814479875

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Automorphic Forms And Shimura Varieties Of Pgsp(2) by Flicker Yuval Z Pdf

The area of automorphic representations is a natural continuation of studies in the 19th and 20th centuries on number theory and modular forms. A guiding principle is a reciprocity law relating infinite dimensional automorphic representations with finite dimensional Galois representations. Simple relations on the Galois side reflect deep relations on the automorphic side, called “liftings.' This in-depth book concentrates on an initial example of the lifting, from a rank 2 symplectic group PGSp(2) to PGL(4), reflecting the natural embedding of Sp(2,≤) in SL(4, ≤). It develops the technique of comparing twisted and stabilized trace formulae. It gives a detailed classification of the automorphic and admissible representation of the rank two symplectic PGSp(2) by means of a definition of packets and quasi-packets, using character relations and trace formulae identities. It also shows multiplicity one and rigidity theorems for the discrete spectrum.Applications include the study of the decomposition of the cohomology of an associated Shimura variety, thereby linking Galois representations to geometric automorphic representations.To put these results in a general context, the book concludes with a technical introduction to Langlands' program in the area of automorphic representations. It includes a proof of known cases of Artin's conjecture.

Author : David Loeffler,Sarah Livia Zerbes
Publisher : Springer
Page : 492 pages
File Size : 43,5 Mb
Release : 2017-01-15
Category : Mathematics
ISBN : 9783319450322

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Elliptic Curves, Modular Forms and Iwasawa Theory by David Loeffler,Sarah Livia Zerbes Pdf

Celebrating one of the leading figures in contemporary number theory – John H. Coates – on the occasion of his 70th birthday, this collection of contributions covers a range of topics in number theory, concentrating on the arithmetic of elliptic curves, modular forms, and Galois representations. Several of the contributions in this volume were presented at the conference Elliptic Curves, Modular Forms and Iwasawa Theory, held in honour of the 70th birthday of John Coates in Cambridge, March 25-27, 2015. The main unifying theme is Iwasawa theory, a field that John Coates himself has done much to create. This collection is indispensable reading for researchers in Iwasawa theory, and is interesting and valuable for those in many related fields.

Author : Yuval Zvi Flicker
Publisher : World Scientific
Page : 499 pages
File Size : 52,7 Mb
Release : 2006
Category : Political Science
ISBN : 9789812568038

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Automorphic Representations of Low Rank Groups by Yuval Zvi Flicker Pdf

The area of automorphic representations is a natural continuation of studies in number theory and modular forms. A guiding principle is a reciprocity law relating the infinite dimensional automorphic representations with finite dimensional Galois representations. Simple relations on the Galois side reflect deep relations on the automorphic side, called ?liftings?. This book concentrates on two initial examples: the symmetric square lifting from SL(2) to PGL(3), reflecting the 3-dimensional representation of PGL(2) in SL(3); and basechange from the unitary group U(3, E/F) to GL(3, E), [E: F] = 2.The book develops the technique of comparison of twisted and stabilized trace formulae and considers the ?Fundamental Lemma? on orbital integrals of spherical functions. Comparison of trace formulae is simplified using ?regular? functions and the ?lifting? is stated and proved by means of character relations.This permits an intrinsic definition of partition of the automorphic representations of SL(2) into packets, and a definition of packets for U(3), a proof of multiplicity one theorem and rigidity theorem for SL(2) and for U(3), a determination of the self-contragredient representations of PGL(3) and those on GL(3, E) fixed by transpose-inverse-bar. In particular, the multiplicity one theorem is new and recent.There are applications to construction of Galois representations by explicit decomposition of the cohomology of Shimura varieties of U(3) using Deligne's (proven) conjecture on the fixed point formula.

Author : Yuval Z. Flicker
Publisher : Birkhäuser
Page : 567 pages
File Size : 46,8 Mb
Release : 2016-09-14
Category : Mathematics
ISBN : 9783319315935

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Arthur's Invariant Trace Formula and Comparison of Inner Forms by Yuval Z. Flicker Pdf

This monograph provides an accessible and comprehensive introduction to James Arthur’s invariant trace formula, a crucial tool in the theory of automorphic representations. It synthesizes two decades of Arthur’s research and writing into one volume, treating a highly detailed and often difficult subject in a clearer and more uniform manner without sacrificing any technical details. The book begins with a brief overview of Arthur’s work and a proof of the correspondence between GL(n) and its inner forms in general. Subsequent chapters develop the invariant trace formula in a form fit for applications, starting with Arthur’s proof of the basic, non-invariant trace formula, followed by a study of the non-invariance of the terms in the basic trace formula, and, finally, an in-depth look at the development of the invariant formula. The final chapter illustrates the use of the formula by comparing it for G’ = GL(n) and its inner form G and for functions with matching orbital integrals.bribr/i/idiviiArthur’s Invariant Trace Formula and Comparison of Inner Forms/div

Author : Gebhard Böckle,Richard Pink
Publisher : European Mathematical Society
Page : 200 pages
File Size : 48,8 Mb
Release : 2009
Category : Mathematics
ISBN : 3037190744

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Cohomological Theory of Crystals Over Function Fields by Gebhard Böckle,Richard Pink Pdf

This book develops a new cohomological theory for schemes in positive characteristic $p$ and it applies this theory to give a purely algebraic proof of a conjecture of Goss on the rationality of certain $L$-functions arising in the arithmetic of function fields. These $L$-functions are power series over a certain ring $A$, associated to any family of Drinfeld $A$-modules or, more generally, of $A$-motives on a variety of finite type over the finite field $\mathbb{F}_p$. By analogy to the Weil conjecture, Goss conjectured that these $L$-functions are in fact rational functions. In 1996 Taguchi and Wan gave a first proof of Goss's conjecture by analytic methods a la Dwork. The present text introduces $A$-crystals, which can be viewed as generalizations of families of $A$-motives, and studies their cohomology. While $A$-crystals are defined in terms of coherent sheaves together with a Frobenius map, in many ways they actually behave like constructible etale sheaves. A central result is a Lefschetz trace formula for $L$-functions of $A$-crystals, from which the rationality of these $L$-functions is immediate. Beyond its application to Goss's $L$-functions, the theory of $A$-crystals is closely related to the work of Emerton and Kisin on unit root $F$-crystals, and it is essential in an Eichler - Shimura type isomorphism for Drinfeld modular forms as constructed by the first author. The book is intended for researchers and advanced graduate students interested in the arithmetic of function fields and/or cohomology theories for varieties in positive characteristic. It assumes a good working knowledge in algebraic geometry as well as familiarity with homological algebra and derived categories, as provided by standard textbooks. Beyond that the presentation is largely self contained.

Author : Hossein Movasati
Publisher : World Scientific
Page : 323 pages
File Size : 53,7 Mb
Release : 2021-10-12
Category : Mathematics
ISBN : 9789811238697

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Modular And Automorphic Forms & Beyond by Hossein Movasati Pdf

The guiding principle in this monograph is to develop a new theory of modular forms which encompasses most of the available theory of modular forms in the literature, such as those for congruence groups, Siegel and Hilbert modular forms, many types of automorphic forms on Hermitian symmetric domains, Calabi-Yau modular forms, with its examples such as Yukawa couplings and topological string partition functions, and even go beyond all these cases. Its main ingredient is the so-called 'Gauss-Manin connection in disguise'.

Author : Armand Borel,W. Casselman,American Mathematical Society
Publisher : American Mathematical Soc.
Page : 394 pages
File Size : 54,8 Mb
Release : 1979-06-30
Category : Mathematics
ISBN : 9780821814376

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Automorphic Forms, Representations and $L$-Functions by Armand Borel,W. Casselman,American Mathematical Society Pdf

Part 2 contains sections on Automorphic representations and $L$-functions, Arithmetical algebraic geometry and $L$-functions

Author : Martin L. Brown
Publisher : Springer
Page : 518 pages
File Size : 51,7 Mb
Release : 2004-08-30
Category : Mathematics
ISBN : 9783540444756

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Heegner Modules and Elliptic Curves by Martin L. Brown Pdf

Heegner points on both modular curves and elliptic curves over global fields of any characteristic form the topic of this research monograph. The Heegner module of an elliptic curve is an original concept introduced in this text. The computation of the cohomology of the Heegner module is the main technical result and is applied to prove the Tate conjecture for a class of elliptic surfaces over finite fields, this conjecture is equivalent to the Birch and Swinnerton-Dyer conjecture for the corresponding elliptic curves over global fields.

Author : Gérard Laumon
Publisher : Cambridge University Press
Page : 362 pages
File Size : 52,7 Mb
Release : 1996
Category : Mathematics
ISBN : 9780521470605

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Cohomology of Drinfeld Modular Varieties, Part 1, Geometry, Counting of Points and Local Harmonic Analysis by Gérard Laumon Pdf

Originally published in 1995, Cohomology of Drinfeld Modular Varieties aimed to provide an introduction, in two volumes, both to this subject and to the Langlands correspondence for function fields. These varieties are the analogues for function fields of the Shimura varieties over number fields. The Langlands correspondence is a conjectured link between automorphic forms and Galois representations over a global field. By analogy with the number-theoretic case, one expects to establish the conjecture for function fields by studying the cohomology of Drinfeld modular varieties, which has been done by Drinfeld himself for the rank two case. The present volume is devoted to the geometry of these varieties, and to the local harmonic analysis needed to compute their cohomology. Though the author considers only the simpler case of function rather than number fields, many important features of the number field case can be illustrated.