Multiplicity one, local and global conjugacy springerlink. Automorphisms of the spaces of automorphic forms 67 11. Within the automorphic lfunctions a special class can be singled out, the class of standard lfunctions, which generalize the hecke functions and for which the analytic continuation and. Lfunctions and automorphic representations 5 where nis a positive integer called the modulus of one says that. Automorphic representations of gsp4 by james arthur 1. This book introduces the reader to the subject and in particular to elliptic modular forms with emphasis on their numbertheoretical aspects. The book features extensive foundational material on the representation theory of gl1 and gl2 over local fields, the theory of automorphic representations, l functions and advanced topics such as the langlands conjectures, the weil representation, the rankinselberg method and the triple l function, examining this subject matter from many. Automorphic forms and representations by daniel bump. Representation theory and automorphic forms american. Measure theory and pseudo almost automorphic function. Automorphic forms, l functions and number theory march 1216 three introductory lectures. Introduction to the arithmetic theory of automorphic functions. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them.
Representation theory and automorphic forms springerlink. The early history of its development has been described in. Taking periods is the classical technique to study automorphic functions. Automorphic functions and automorphic representations. Stanton abstract let g be a connected, real, semisimple lie group contained in its complexi. For each absolute value v on f, fv denotes the completion of f with respect to v, and if v is. Lfunctions and automorphic representations department of. The theory of automorphic functions, or fuchsian functions as poincare called them, is a fruitful result of using complex function theory in the analysis of linear ordinary differential equations odes. An automorphic form, constructed as a nonholomorphic. String theory is a very large area of research with many subjects beyond the scope of this thesis. This book, which is volume 17 in the amslms history series, contains a small book by jacques hadamard on the connections between noneuclidean geometry and the theory of automorphic functions. The theory of automorphic forms is playing increasingly important roles in several branches of mathematics, even in physics, and is almost ubiquitous in number theory.
Lectures on automorphic l functions james arthur and stephen gelbart preface this article follows the format of five lectures that we gave on automorphic l functions. Model companions of theories with an automorphism kikyo, hirotaka, journal of symbolic logic, 2000. This volume addresses the interplay between representation theory and automorphic forms. Goro shimura, introduction to the arithmetic theory of automorphic functions larry joel goldstein. Lfunctions, and number theory math user home pages. New developments and applications article pdf available in fuel and energy abstracts 754 march 2012 with 270 reads how we measure reads. Representation theory and automorphic functions izrail. We construct a kcg double coset domain in gc, and we show that the action of g on the. Solomon friedberg boston college automorphic forms icerm, jan.
Measure theory and pseudo almost automorphic functions. Goro shimuras 1971 monograph, introduction to the arithmetic theory of automorphic functions, published originally by iwanami shoten together with princeton university press, and now reissued in paperback by princeton, is one of the most important books in the subject. An introduction to the theory of automorphic functions by ford, lester r. An introduction to the theory of automorphic functions. Our goal is to formulate a theorem that is part of a recent classi cation of automorphic representations of orthogonal and symplectic groups. Other readers will always be interested in your opinion of the books youve read. The twentyfifth ams summer research institute was devoted to automorphic forms, representations and lfunctions. If f is a function on x then we denote by suppf the closure of. In harmonic analysis and number theory, an automorphic form is a wellbehaved function from a topological group g to the complex numbers or complex vector space which is invariant under the action of a discrete subgroup. Gelfand, graev, piatetskiishapiro, representation theory and automorphic functions. Automorphic forms, lfunctions and number theory march 1216. The lectures were intended to be a brief introduction for number theorists to some of the main ideas in the subject. It was held at oregon state university, corvallis, from july 11 to august 5, 1977, and was financed by a grant from the national science foundation. Pdf the rankinselberg method for automorphic distributions.
To place it in perspective, we devote much of the paper to a historical introduction to the langlands program. The first systematic theory of generalized functions also known as distributions was created in the early 1950s, although some aspects were developed much earlier, most notably in the definition of the greens function in mathematics and in the work of paul dirac on quantum electrodynamics in physics. For example, the space of constant functions is a discrete automorphic representa tion, because g has finite volume for g semisimple. In addition to the results discussed above, methods in the theory of automorphic functions are important in the study of moduli varieties for objects such as algebraic curves and abelian varieties. The results are part of a project a3 on the automorphic representations of general classical groups. Galois representations and automorphic forms mastermath. Gunning and leon ehrenpreis, editors, theta functions bowdoin 1987. Automorphic forms, lfunctions and number theory march 12. 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. Here is the official catalog description from cambridge. In this work, we establish a new concept of weighted pseudo almost automorphic functions using the measure theory. The reciprocitylaw at cmpoints and rationality of automorphic forms 58 10. Representation theory and automorphic forms toshiyuki. Noneuclidean geometry in the theory of automorphic functions.
One is generating number elds with extreme class numbers. Decomposition of representations into tensor products 179 190. Automorphic l functions and their applications to number theory jaehyun peter cho doctor of philosophy graduate department of mathematics university of toronto 2012 the main part of the thesis is applications of the strong artin conjecture to number theory. We study these problems in a setting related to the langlands lfunctions lps. The basic example is g acting on l2g by left or right translation. Compositions of pseudo almost automorphic functions via measure theory and applications fan, zhenbin, dong, qixiang, and li, gang, journal of integral equations and applications, 2016.
We recall the standard connection of the above setting with representation theory. It was followed by two volumes on elliptic modular functions by klein and fricke and two more volumes on automorphic functions also by klein and fricke. This paper is an expository account of our 1976 monograph 6 on scattering theory for automorphic functions. The theory of this work generalizes the classical results on weighted pseudo almost periodic and automorphic functions. One could describe an automorphic representation of. Automorphic forms, lfunctions and number theory march 1216 three introductory lectures e. Often the space is a complex manifold and the group is a discrete group. It is cause for joy and celebration that the sources subseries of the amslms history of mathematics series continues to grow. A main goal of the institute was the discussion of the l functions attached to automorphic forms on, or automorphic representations of, reductive groups, the local and global problems pertaining to them, and of their relations with the l functions of algebraic number theory and algebraic geometry, such as artin l.
Hundley automorphic representations and lfunctions for the general linear group, i d. Hundley automorphic representations and lfunctions for the general linear group, ii. L functions and automorphic representations james arthur abstract. Representation theory relates to number theory through the langlands program, which conjecturally connects algebraic extensions of number fields to automorphic representations and l functions. Publication date 1915 topics automorphic functions. Automorphic functions are also of importance in number theory. Estimates of automorphic functions school of mathematical sciences. Introduction to the arithmetic theory of automorphic. These four classic books are vast generalizations of the first volume and contain the highly original works of poincare and klein on automorphic forms. An introduction to automorphic representations duke mathematics. Here are two fairly old books that explain and exploit representation theory behind the theory of theta functions and automorphic forms neither assuming nor using algebraic geometry and commutative algebra in a serious way.
Automorphic forms, representations, and l functions. Introductory lectures on automorphic forms ucsd mathematics. How is representation theory used in modularautomorphic. Representation theory and automorphic functions generalized functions, vol 6 9780122795060. The issue of local and global conjugacy is closely related to the multiplicity one property in representation theory and the langlands program. This book, published by cambridge university press, is now shipping. We present new results on weighted ergodic functions like completeness and composition theorems. Automorphic forms, representation theory and arithmetic. Krein scattering theory for automorphic functions by peter d.
Langlands beyond endoscopy proposal for establishing functoriality motivates interesting and concrete problems in the representation theory of algebraic groups. Multiple dirichlet series, lfunctions and automorphic forms. This volume addresses the interplay between representation theory and. On tensor third lfunctions of automorphic representations of glnpafq heekyoung hahn abstract.
In chapter2, some fundamental concepts of string theory are introduced followed by a more detailed motivation for studying automorphic forms in string theory discussing nonperturbative contributions to fourgraviton scatteringamplitudesbasedon4. In this article we give first families of connected instances for so2n where the multiplicity one fails in both aspects of representation theory and automorphic forms with certain assumptions on the langlands functoriality. There has recently been much interest, if not a tremendous amount of progress, in the. The study of left invariant functions on g is of interest.
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