On the Stabilization of the Trace Formula
Author: Laurent Clozel
Publisher: International Pressof Boston Incorporated
Published: 2011
Total Pages: 527
ISBN-13: 9781571462275
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Harmonic Analysis, the Trace Formula, and Shimura Varieties
Author: Clay Mathematics Institute. Summer School
Publisher: American Mathematical Soc.
Published: 2005
Total Pages: 708
ISBN-13: 9780821838440
DOWNLOAD EBOOKLanglands program proposes fundamental relations that tie arithmetic information from number theory and algebraic geometry with analytic information from harmonic analysis and group representations. This title intends to provide an entry point into this exciting and challenging field.
Shimura Varieties
Author: Thomas Haines
Publisher: Cambridge University Press
Published: 2020-02-20
Total Pages: 341
ISBN-13: 1108704867
DOWNLOAD EBOOKThis volume forms the sequel to "On the stabilization of the trace formula", published by International Press of Boston, Inc., 2011
Relative Trace Formulas
Author: Werner Müller
Publisher: Springer Nature
Published: 2021-05-18
Total Pages: 438
ISBN-13: 3030685063
DOWNLOAD EBOOKA series of three symposia took place on the topic of trace formulas, each with an accompanying proceedings volume. The present volume is the third and final in this series and focuses on relative trace formulas in relation to special values of L-functions, integral representations, arithmetic cycles, theta correspondence and branching laws. The first volume focused on Arthur’s trace formula, and the second volume focused on methods from algebraic geometry and representation theory. The three proceedings volumes have provided a snapshot of some of the current research, in the hope of stimulating further research on these topics. The collegial format of the symposia allowed a homogeneous set of experts to isolate key difficulties going forward and to collectively assess the feasibility of diverse approaches.
Level One Algebraic Cusp Forms of Classical Groups of Small Rank
Author: Gaëtan Chenevier
Publisher: American Mathematical Soc.
Published: 2015-08-21
Total Pages: 134
ISBN-13: 147041094X
DOWNLOAD EBOOKThe authors determine the number of level 1, polarized, algebraic regular, cuspidal automorphic representations of GLn over Q of any given infinitesimal character, for essentially all n≤8. For this, they compute the dimensions of spaces of level 1 automorphic forms for certain semisimple Z-forms of the compact groups SO7, SO8, SO9 (and G2) and determine Arthur's endoscopic partition of these spaces in all cases. They also give applications to the 121 even lattices of rank 25 and determinant 2 found by Borcherds, to level one self-dual automorphic representations of GLn with trivial infinitesimal character, and to vector valued Siegel modular forms of genus 3. A part of the authors' results are conditional to certain expected results in the theory of twisted endoscopy.
On the Cohomology of Certain Non-Compact Shimura Varieties
Author: Sophie Morel
Publisher: Princeton University Press
Published: 2010-01-31
Total Pages: 230
ISBN-13: 0691142920
DOWNLOAD EBOOKThis 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) of finite adelic points of G. The second action can be studied on the set of complex points of the Shimura variety. In this book, Sophie Morel identifies the Galois action--at good places--on the G(Af)-isotypical components of the cohomology. Morel uses the method developed by Langlands, Ihara, and Kottwitz, which is to compare the Grothendieck-Lefschetz fixed point formula and the Arthur-Selberg trace formula. The first problem, that of applying the fixed point formula to the intersection cohomology, is geometric in nature and is the object of the first chapter, which builds on Morel's previous work. She then turns to the group-theoretical problem of comparing these results with the trace formula, when G is a unitary group over Q. Applications are then given. In particular, the Galois representation on a G(Af)-isotypical component of the cohomology is identified at almost all places, modulo a non-explicit multiplicity. Morel also gives some results on base change from unitary groups to general linear groups.
Shimura Varieties
Author: Thomas Haines
Publisher: Cambridge University Press
Published: 2020-02-20
Total Pages: 341
ISBN-13: 1108632068
DOWNLOAD EBOOKThis is the second volume of a series of mainly expository articles on the arithmetic theory of automorphic forms. It forms a sequel to On the Stabilization of the Trace Formula published in 2011. The books are intended primarily for two groups of readers: those interested in the structure of automorphic forms on reductive groups over number fields, and specifically in qualitative information on multiplicities of automorphic representations; and those interested in the classification of I-adic representations of Galois groups of number fields. Langlands' conjectures elaborate on the notion that these two problems overlap considerably. These volumes present convincing evidence supporting this, clearly and succinctly enough that readers can pass with minimal effort between the two points of view. Over a decade's worth of progress toward the stabilization of the Arthur-Selberg trace formula, culminating in Ngo Bau Chau's proof of the Fundamental Lemma, makes this series timely.
Automorphic Forms Beyond $mathrm {GL}_2$
Author: Ellen Elizabeth Eischen
Publisher: American Mathematical Society
Published: 2024-03-26
Total Pages: 199
ISBN-13: 1470474921
DOWNLOAD EBOOKThe Langlands program has been a very active and central field in mathematics ever since its conception over 50 years ago. It connects number theory, representation theory and arithmetic geometry, and other fields in a profound way. There are nevertheless very few expository accounts beyond the GL(2) case. This book features expository accounts of several topics on automorphic forms on higher rank groups, including rationality questions on unitary group, theta lifts and their applications to Arthur's conjectures, quaternionic modular forms, and automorphic forms over functions fields and their applications to inverse Galois problems. It is based on the lecture notes prepared for the twenty-fifth Arizona Winter School on “Automorphic Forms beyond GL(2)”, held March 5–9, 2022, at the University of Arizona in Tucson. The speakers were Ellen Eischen, Wee Teck Gan, Aaron Pollack, and Zhiwei Yun. The exposition of the book is in a style accessible to students entering the field. Advanced graduate students as well as researchers will find this a valuable introduction to various important and very active research areas.
The Genesis of the Langlands Program
Author: Julia Mueller
Publisher: Cambridge University Press
Published: 2021-08-05
Total Pages: 452
ISBN-13: 1108619959
DOWNLOAD EBOOKRobert Langlands formulated his celebrated conjectures, initiating the Langlands Program, at the age of 31, profoundly changing the landscape of mathematics. Langlands, recipient of the Abel Prize, is famous for his insight in discovering links among seemingly dissimilar objects, leading to astounding results. This book is uniquely designed to serve a wide range of mathematicians and advanced students, showcasing Langlands' unique creativity and guiding readers through the areas of Langlands' work that are generally regarded as technical and difficult to penetrate. Part 1 features non-technical personal reflections, including Langlands' own words describing how and why he was led to formulate his conjectures. Part 2 includes survey articles of Langlands' early work that led to his conjectures, and centers on his principle of functoriality and foundational work on the Eisenstein series, and is accessible to mathematicians from other fields. Part 3 describes some of Langlands' contributions to mathematical physics.
Automorphic Forms and Even Unimodular Lattices
Author: Gaëtan Chenevier
Publisher: Springer
Published: 2019-02-28
Total Pages: 428
ISBN-13: 3319958917
DOWNLOAD EBOOKThis book includes a self-contained approach of the general theory of quadratic forms and integral Euclidean lattices, as well as a presentation of the theory of automorphic forms and Langlands' conjectures, ranging from the first definitions to the recent and deep classification results due to James Arthur. Its connecting thread is a question about lattices of rank 24: the problem of p-neighborhoods between Niemeier lattices. This question, whose expression is quite elementary, is in fact very natural from the automorphic point of view, and turns out to be surprisingly intriguing. We explain how the new advances in the Langlands program mentioned above pave the way for a solution. This study proves to be very rich, leading us to classical themes such as theta series, Siegel modular forms, the triality principle, L-functions and congruences between Galois representations. This monograph is intended for any mathematician with an interest in Euclidean lattices, automorphic forms or number theory. A large part of it is meant to be accessible to non-specialists.
