Lfunctions and automorphic representations james arthur abstract. Dihua jiang professor automorphic forms, l functions, number theory, harmonic analysis, representation theory. 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. Qua composition, shimura states that the raison detre of his book is the treatment of complex multiplication of elliptic or elliptic modular functions. Automorphic lfunctions 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. Shimura introduction to the arithmetic theory of automorphic func. Number theory and automorphic forms school of mathematics. Introduction to the arithmetic theory of automorphic functions publications of the mathematical society of japan 11. Weitz, introduction to cardinal arithmetic burke, maxim r. Automorphic forms, lfunctions and number theory march 12.
Arithmetic functions today arithmetic functions, the mobius. In the seminar the speaker mentioned that this is an automorphic form. 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. We study these problems in a setting related to the langlands lfunctions lps. I will concentrate on the case of unitary groups since this seems the most relevant for current applications. In the present paper, we will concentrate on discussing some basic analytical and arith. Msri model theory, arithmetic geometry and number theory. Colloquium was to discuss recent achievements in the theory of automorphic forms of one and several variables, representation theory with special reference to the interplay between these and number theory, e. Introduction to functions mctyintrofns20091 a function is a rule which operates on one number to give another number. The theory of automorphic forms is playing increasingly important roles in several branches of mathematics, even in physics, and is almost. Representation theory and automorphic functions izrail. The reciprocitylaw at cmpoints and rationality of automorphic forms 58 10. Solomon friedberg boston college automorphic forms icerm, jan.
Lfunctions, and number theory math user home pages. Ford, an introduction to the theory of automorphic functions emch, arnold, bulletin of the american mathematical society, 1916. Introduction to arithmetic theory of automorphic functions. Automorphic forms are a generalization of the idea of periodic functions in euclidean space to general topological groups. To place it in perspective, we devote much of the paper. Indeed, there are some very simple multiplicative functions. Kaiwen lan associate professor number theory, automorphic forms, shimura varieties and related topics in arithmetric geometry. An example of an arithmetic function is the divisor function. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. Lfunctions and automorphic forms heidelberg university. Arithmetic with satisfaction cain, james, notre dame journal of formal logic, 1995.
Part i the global lfunction is then given as a formal product if the global lfunction is to have interesting arithmetic properties, one needs to assume that v is automorphic. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. This book introduces the reader to the subject and in particular to elliptic modular forms with emphasis on their numbertheoretical aspects. The fruitful interaction between automorphic forms and arithmetic has produced a cornucopia of fundamental results, including such milestones as the shimurataniyama conjecture, the birch and swinnertondyer conjecturefor elliptic curves of low analytic rank, the serre, fontainemazur, and artin conjecturesfor odd twodimensional galois representations, and the satotate conjectures. Written by one of the leading experts, venerable grandmasters, and most active contributors \\ldots\ in the arithmetic theory of automorphic forms \\ldots\ the new material included here is mainly the outcome of his extensive work \\ldots\ over the last eight years \\ldots\ a very careful, detailed introduction to the subject \\ldots\ this monograph is an important. Automorphic forms, representation theory and arithmetic. Goro shimura, introduction to the arithmetic theory of automorphic. Automorphic representations and galois representations. Number theory question with automorphic numbers mathematics. In mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient space. Buy introduction to the arithmetic theory of automorphic functions publications of the mathematical society of japan, vol.
Ford, an introduction to the theory of automorphic functions emch, arnold, bulletin of the american. Often the space is a complex manifold and the group is a discrete group. Publications of the mathematical society of japan, no. Automorphic forms, lfunctions and number theory march 1216. Workshop automorphic galois representations, lfunctions. Construction of automorphic galois representations haruzo hida. Whenever this is possible one is able to prove rationality results for ratios of critical values of certain automorphic l functions such as in the examples listed above. Publication date 1915 topics automorphic functions publisher london g. Formulas for divisors of a function and form are proved and their consequences analyzed. Those, notably the notions of conductor and of primitivity, and the link with class.
The study of left invariant functions on g is of interest. Introduction to the arithmetic theory of automorphic. Modern analysis of automorphic forms by example current version is my 485page, in 8. Modular forms are automorphic forms defined over the groups sl2, r or psl2, r with the discrete subgroup being the modular group, or one of its congruence subgroups. An introduction to the theory of automorphic functions by ford, lester r. It is unimaginable that a number theorist, be he a. This thesis represents an introduction to the study of elementary number theory from a point of view that is exquisitely a mix of algebraic, analytical and combinatorial elements and concepts. Apr 05, 2018 once this context is in place, one may then try to view langlandss constant term theorem, which sees ratios of products of automorphic l functions, in terms of maps in cohomology. Pdf introduction to the arithmetic theory of automorphic functions.
As subgroups of the group of automorphism of the upper half plane, fuchsian groups operate on the upper half plane as well. Finally, one must mention the application of automorphic functions to the study of ordinary differential equations in a complex domain 12 and in the construction of solutions of algebraic. Automorphic forms, representation theory and arithmetic papers presented at the bombay colloquium 1979. Modular forms and arithmetic geometry kudla, stephen s. Dihua jiang professor automorphic forms, lfunctions, number theory, harmonic analysis, representation theory. Automorphic lfunctions and their applications to number. 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. This book introduces the reader to the subject and in particular to elliptic modular forms with. This unit explains how to see whether a given rule describes a valid function, and introduces some of the mathematical terms associated with functions. The most important arithmetic functions in number theory are the multiplicative functions, those which satisfy m,n1. Of course in analysis most interesting functions are not just polynomials. University of copenhagen, department of mathematical sciences. Arithmetic and hyperbolic structures in string theory.
Chapter two develops automorphic functions and forms via the poincare series. Langlands beyond endoscopy proposal for establishing functoriality motivates interesting and concrete problems in the representation theory of algebraic groups. Introduction to the arithmetic theory of automorphic functions. Given g, i am going to introduce a complex analytic group g. Jan hendrik bruinier darmstadt winfried kohnen heidelberg list of speakers.
Bell journal of the indian mathematical society, october, 1928, wherein he pointed out that he had established the existence of the inverse function, for a wider class of functions than the multiplicative, and gave a general. On tensor third lfunctions of automorphic representations of glnpafq heekyoung hahn abstract. Since then ive been trying to find out what an automorphic form is but a search for a book on the material usually yields results such as. It is also beautifully structured and very wellwritten, if compactly. Very roughly, this is the arithmetic analogue of the analytical problem expressing a realvalued function fx as a combination of simple functions like xk or cosnx, sinnx. During this academic year, henryk iwaniec and peter sarnak will be in residence at the institute for advanced study and there will be a program with the purpose to bring together specialists in analytic number theory and specialists in the analytic theory of automorphic forms. Shimura,arithmeticity in the theory of automorphic forms. One is generating number elds with extreme class numbers.
We consider the reproducing kernel function of the theta bargmannfock hilbert space associated to given fullrank lattice and pseudocharacter, and we deal with some of. The workshop and the papers contributed to this volume circle around such topics as the theory of automorphic forms and their \l\ functions, geometry and number theory, covering some of the recent approaches and advances to these subjects. However, not every rule describes a valid function. Workshop automorphic galois representations, lfunctions and. Written by one of the leading experts, venerable grandmasters, and most active contributors \\ldots\ in the arithmetic theory of automorphic forms \\ldots\ the new material included here is mainly the outcome of his extensive work \\ldots\ over the last eight years \\ldots\ a very careful, detailed introduction to the subject \\ldots\ this monograph is an important, comprehensively. Arithmetic functions, lecture notes mit opencourseware.
In mathematics, an automorphic lfunction is a function ls. View the article pdf and any associated supplements and figures for a period of 48 hours. Ford, an introduction to the theory of automorphic functions emch, arnold, bulletin of the american mathematical. The final chapter is devoted to the connection between automorphic function theory and riemann surface theory, concluding with some applications of riemannroch theorem. Introduction to the arithmetic theory of automorphic functions publications of the mathematical society of japan 11 goro shimura 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.
The workshop spectral theory, automorphic forms and arithmetic will cover a wide range of topics related to various analytic aspects of number theory with emphasis on spectral theory, automorphic forms and arithmetic. A short course in automorphic functions dover books on. Analytic and arithmetic properties of the g,cautomorphic reproducing kernel function a. Our goal is to formulate a theorem that is part of a recent classi cation of automorphic representations of orthogonal and symplectic groups. For each absolute value v on f, fv denotes the completion of f with respect to v, and if v is. Arithmetic theta lifts and the arithmetic gangrossprasad conjecture for unitary groups xue, hang, duke mathematical journal, 2019. Arithmetic functions are different from typical functions in that they cannot usually be described by simple formulas, so they are often evaluated in terms of their average or asymptotic behavior. Introduction to arithmetic theory of automorphic functions pdf. This talk will be an exposition of a circle of ideas that concerns the cohomology of arithmetic groups and the special values of certain automorphic lfunctions. I will again try to sketch the construction, what is known and how far we are from a full theory. Analytic theory of automorphic forms and lfunctions ias.
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