This volume addresses the interplay between representation theory and automorphic forms. Given a global eld fand a reductive algebraic group gover f, then an automorphic representation of g is a g. We provide an introduction to the theory of eisenstein series and automorphic forms on real simple lie groups g, emphasising the role of representation theory. Kowalski, section 3 of automorphic forms, lfunctions and number theory march 1216 three introductory lectures dorian goldfeld, joseph hundley, chapter 2 of automorphic representations and lfunctions for the general linear group, cambridge studies in advanced mathematics 129, 2011 pdf. The book features extensive foundational material on the representation theory of gl 1 and gl 2 over local fields, the theory of automorphic representations, lfunctions and advanced topics such as the langlands conjectures, the weil representation, the rankinselberg method and the triple lfunction, and examines this subject matter from many. On tensor third lfunctions of automorphic representations of glnpafq heekyoung hahn abstract. This book, published by cambridge university press, is now shipping. Representation theory, automorphic forms, and complex geometry a conference in honor of the 70th birthday of wilfried schmid harvard university, science center hall c, may 2023, 20. Using these automorphic forms, one can construct the automorphic representation associated to f. Representation theory relates to number theory through the langlands program, which conjecturally connects algebraic extensions of number fields to automorphic representations and lfunctions. We have a satisfactory notion of automorphic form when r c.
Automorphic forms, representations, and combinatorics. Langlands beyond endoscopy proposal for establishing functoriality motivates interesting and concrete problems in the representation theory of algebraic groups. We study these problems in a setting related to the langlands lfunctions lps. Representation theory and automorphic forms toshiyuki. Conjecture langlands for any continuous representation of the galois group there is an automorphic representation tt 6 iig, neces sarily unique, such that ova cr,r for all v outside some finite set s 2 sm. To place it in perspective, we devote much of the paper to a historical introduction to the langlands program. The subsequent notion of an automorphic representation has proved of great technical value when dealing with g an algebraic group, treated as an adelic algebraic group. Automorphic representations and galois representations. Pdf fourier coefficients and small automorphic representations. We shall begin with a brief description of the basic results in this theory, and then give a reformulation using representation the ory. Galois representations86 references 86 introduction the goal of this course is to introduce and study automorphic representations. The conjecture represents a fundamental problem in number theory.
G be the center, so that z is isomorphic to the multiplicative group. A guiding principle is a reciprocity law relating the infinite dimensional automorphic representations with finite dimensional galois representations. Representation theory of the latter group naturally leads to the study of automorphic functions and related numbertheoretic problems. Knapp, editors, representation theory and automorphic forms international centre for mathematical sciences. It does not completely include the automorphic form idea introduced above, in that the adelic approach is a way of dealing with the whole family of congruence subgroups at once. The representation theory of glpnqis relatively simple, and quite well understood now, thanks to the work of a number of mathematicians over the past forty years.
We give two examples that will later turn out to be known special cases of the langlands correspondence, namely gausss quadratic reciprocity theorem and the modularity theorem of wiles et al. The basic example is g acting on l2g by left or right translation. There are several different classes of groups, associative algebras and lie algebras, and their representation theories all have an individual flavour. The latter control the embedding of a model of a unitary representation into spaces ofautomorphicfunctions,andwere earlier studiedby ehrenpreis. Workshop automorphic galois representations, lfunctions and. The area of automorphic representations is a natural continuation of studies in number theory and modular forms. Workshop automorphic galois representations, lfunctions.
For representation theory of finite groups, the classic is serres linear representations of finite groups. Shahidis paper intertwining operators, lfunctions, and representation theory, notes of the eleventh kaist mathematics workshop 1996, taejon, korea. Automorphic forms, representation theory and arithmetic. These conjectures led to the theory of modular forms. Reference for automorphic form via representation theory.
The above theory of hecke works beautifully for modular forms with respect to sl2. Here is the official catalog description from cambridge. Decomposition of representations into tensor products 179 190. If g is a topological group, then a unitary representation of g is an isometric action of g on a hilbert space h so that the action map g. The conference automorphic forms, representations, and combinatorics will take place at stanford university from august 16, 2012. Galois representation there are huge differences between the local representation at p6.
Wallach 1 orbital integrals and the harishchandra transform. On the notion of an automorphic representation the. Classifying automorphic representations mathematical institute. Introductory lectures on automorphic forms lectures for the european school of group theory july, 2001, luminy, france by nolan r.
This reformulation leads to a vast generalization of the theory. Representation theory depends upon the type of algebraic object being represented. The purpose of the colloquium was to discuss recent achievements in the theory of automorphic forms of one and several variables, representation theory with. Automorphy for some ladic lifts of automorphic mod l galois. The article that marc palm is mentioning has been mention to me by other people. Representation theory university of california, berkeley. Eisenstein series and automorphic representations the moduli space of super riemann surfaces since one is dealing with superstring theory and it is known that this integral cannot be reduced in a simple manner to an integral over ordinary riemann surfaces for h5 111. To place it in perspective, we devote much of the paper.
Spg reading online ajcc cancer staging manual 7th edition pdf free pdf bookboon reading online ajcc cancer staging manual 7th edi. We note that the general langlands correspondence is still largely. The invited papers, written by leading mathematicians, track recent progress in the ever expanding fields of representation theory and automorphic forms, and their association with number theory and. Automorphic forms and representations by daniel bump. An introduction to automorphic representations duke mathematics. Representation theory depends upon the nature of the vector space on which the algebraic object is represented. The third chapter is devoted to the local theory of galois representations. However, no body has ever been able to point out the precise definition or theorem or anything concrete. It is not used in the hecke theory but in the chapter on automorphic forms and quaternion algebras. An automorphic form, constructed as a nonholomorphic. Jun 11, 2014 automorphic representations in form of induced representations with inducing quantum bundle ov er a riemann surface and then use the loop group represen tation construction to realize the. It is good to learn modular forms or elliptic curves first, though most accounts of modular forms dont make the representation theory aspect evident.
Representation theory, automorphic forms, and complex geometry. Given an automorphic form f, roughly speaking, one considers the vector space v. G can be embedded in gl n in such a way that it is stable under transpose. Representation theory and automorphic forms american. The invited papers, written by leading mathematicians, track recent progress in the ever expanding fields of representation theory and automorphic forms, and their association with. Iv automorphic representations 27 basic representation theory of real and padic groups. In the mid1990s wilfried schmid became interested in applying analytic concepts from representation theory to automorphic forms via their boundary value distributions. Aninternationalcolloquiumon automorphic forms, representation theory and arithmetic was held at the tata institute of fundamental research, bombay, from 8 to 15 january 1979. In the language of representation theory one of the fundamental problems of the modern theory of automorphic forms is the following. Database of automorphic lfunctions rutgers university. Theres a lot involved in automorphic forms, and a lot of aspects to come at it from. In this theory, one considers representations of the group algebra a cg of a. Representation theory and automorphic forms springerlink.
I will present hida theory for automorphic forms and its application to the study of the eisenstein ideal. The theory of automorphic forms then shows that ls. Our goal is to formulate a theorem that is part of a recent classi cation of automorphic representations of orthogonal and symplectic groups. Zhang shtukas and the taylor expansion of lfunctions ii. Classifying automorphic representations clay mathematics institute. Galois representations and automorphic forms mastermath. We shall begin with a brief description of the basic results in this theory, and then give a reformulation using representation theory. Daniel bump has been a leading mathematician in automorphic forms, representation theory. Then r arises from a cuspidal automorphic representation. The advantage of elliptic curves is that you will probably see the representation theory sooner. The second conference, entitled representation theory, automorphic forms, and complex geometry, was hosted by harvard university in honor of prof. Jan 07, 2020 the second conference, entitled representation theory, automorphic forms, and complex geometry, was hosted by harvard university in honor of prof. Automorphic forms and automorphic representations wee teck gan references.
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