A valuable resource for mathematicians, the journal provides an international forum for. A simple proof of irreducibility of the generalized igusa. Introduction to the arithmetic theory of automorphic functions paperback ed. It was held at the fields institute in toronto, canada, from june 2 to june 27, 2003. Find all the books, read about the author, and more. For almost 25 years now, hida has been working out that insight. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field. Contributions to automorphic forms, geometry, and number. This book covers the following three topics in a manner accessible to graduate students who have an understanding of algebraic number theory and scheme theoretic algebraic geometry. The clay mathematics institute is organizing a summer school in automorphic forms in june, 2003. Automorphic functions for the whiteheadlinkcomplement. Buy automorphic functions and number theory lecture notes in mathematics on free shipping on qualified orders automorphic functions and number theory lecture notes in mathematics. It is also beautifully structured and very wellwritten, if compactly. However, in large part because of the lack of a suitable text this theory has been difficult for students to learn.
Generating functions qnumber probabilistic number theory computational number theory. It was an essential ingredient of the proof of fermats last theorem. In our situation, d is a hermitian symmetric space of dimension n. Automorphic functions and number theory lecture notes in mathematics, 54 by goro shimura author visit amazons goro shimura page. An introduction to the analytic theory of numbers, raymond g. The main goal of the school was to introduce graduate students and young mathematicians to three broad and interrelated areas in the theory of automorphic forms. Fields institute clay mathematics institute summer school. Theory of n nostalgia, released 01 december 2017 1. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext.
These automorphic functions give embeddings of the orbit spaces of h3 under these groups, and arithmetical characterizations of them. The journal of number theory features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. Shimura curves are the onedimensional shimura varieties. With gis associated an lgroup lg, which is a reductive algebraic group over c that functions in some respects as a dual to g. We construct automorphic functions on the real 3dimensional hyperbolic space h3 for the whiteheadlinkcomplement groupw gl2zi and for a few groups commensurable with w. The coe cients a p a pf for pprime are related to the hecke eigenvalues by t pf a pf. The reciprocitylaw at cmpoints and rationality of automorphic forms 58 10. Number theory or arithmetic or higher arithmetic in older usage is a branch of pure mathematics devoted primarily to the study of the integers and integervalued functions. Weil, varietes abeliennes et courbes algebriques, hermann, paris, 1948. Goro shimura, introduction to the arithmetic theory of automorphic functions larry joel goldstein.
This would bea mammoth task, and limitations oftimeand energyhave considerably reducedthe compassofthis report. Automorphic functions and number theory goro shimura. Automorphic functions are generalizations of trigonometric functions and elliptic functions. The twentyfifth ams summer research institute was devoted to automorphic forms, representations and lfunctions. Number theory, volume 1 by kazuya kato, 9780821808634, available at book depository with free delivery worldwide. Buy automorphic functions and number theory lecture notes in mathematics on free shipping on qualified orders. 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. The new book of prime number records, 3rd edition, p. Generally we speak of the weierstrass function as if tau were constant making it an elliptic function, but it is also possible to speak of the weierstrass function as a laurent series in its. Automorphic functions and number theory lecture notes in mathematics 1968th edition by goro shimura author. Automorphic forms and cohomology theories on shimura curves of small discriminant michael hill. Automorphisms of the spaces of automorphic forms 67 11.
An elementary construction of shimura varieties as moduli of abelian schemes. Math848topics in number theoryshimura varieties and automorphic forms tonghai yang in this topics course, we talk about generalization of the classical modular curves and modular forms, which is about gl 2 theory, to high dimensional analogues induced by shimura in 60s and reformulated by deligne in early 70s. Ben briggs dance with me, woman theory of n remix 11. Milne, canonical models of mixed shimura varieties and automorphic vector bundles, automorphic forms, shimura varieties and functions, perspect.
A short course in automorphic functions, joseph lehner, dover. Contributions to automorphic forms, geometry, and number theory will likely lead to vital interaction among researchers and also help prepare students and other young mathematicians to enter this exciting area of pure mathematics. This book introduces the reader to the subject and in particular to elliptic modular forms with emphasis on their numbertheoretical aspects. When one tries to read about automorphic representation few terms come up more than others namely, 1. In the easiest case, one obtains a transparent formulation by means of roots of unity, which are special values of exponential functions. Modularity of generating functions of special cycles on.
Reciprocity laws of various kinds play a central role in number theory. In his introduction to this book, the author tells of how in 1980 he had the insight that the padic lfunctions attached to automorphic forms on various different algebraic groups should be functions of several variables. Applications of functional analysis and operator theory. For a prime p and a prime power q pk, we write f q for the. Our proofs require a number of tools from arakelov geometry, analytic number theory, galois representations, complexanalytic estimates on shimura curves, automorphic forms, known cases of the colmez conjecture, and results on generalized fermat equations. Roughly speaking, the goal in the study of shimura varieties is to generalize everything that is known about modular curves to all shimura varieties. A similar theory can be developed for special values of elliptic or elliptic modular. In number theory, a shimura variety is a higherdimensional analogue of a modular curve that arises as a quotient variety of a hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over q. We describe some of this theory and its applications. 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. So kevins question should be seen as a program for future collaboration between number theorists and algebraic geometers at least.
The theory of automorphic forms is a cornerstone in modern number theory. This book introduces the reader to the subject and in particular to elliptic modular forms with. Shimura varieties are not algebraic varieties but are families of algebraic varieties. German mathematician carl friedrich gauss 17771855 said, mathematics is the queen of the sciencesand number theory is the queen of mathematics. The books ultimate objective is to prove algebraicity of the critical values of the zeta functions of automorphic forms on unitary and symplectic groups. Download for offline reading, highlight, bookmark or take notes while you read the geometry and cohomology of some simple shimura varieties. It is unimaginable that a number theorist, be he a. Capproximation theory has provided the foundation for many of the most important conceptual breakthroughs and applications of operator algebras. Goro shimura 1930 2019 mactutor history of mathematics. The rings of automorphic forms can then be explicitly determined away from 2 in terms of meromorphic sections of the cotangent bundle. Zink, period spaces for pdivisible groups, annals of mathematics studies 141, princeton university press, 1996, isbn 069102782x. This book introduces the reader to the subject and in particular to elliptic modular forms with emphasis on their number theoretical aspects. Adler, james arthur, don blasius, siegfried boecherer, daniel bump, william casselmann, laurent clozel, james cogdell, laurence corwin, solomon friedberg, masaaki furusawa, benedict gross, thomas hales, joseph harris, michael. Automorphic forms and cohomology theories on shimura curves of small discriminant michael hilla, tyler lawsonb,1 adepartment of mathematics, university of virginia charlottesville, va 2290447 bdepartment of mathematics, university of minnesota minneapolis, mn 55455 abstract we apply luries theorem to produce spectra associated to 1.
Automorphic forms and the periods of abelian varieties. Am151 ebook written by michael harris, richard taylor. It is important to generalize these facts, namely, to investigate in a more general case the relation between automorphic functions with one or more. Abelian varieties with complex multiplication and modular. It had been my intention to survey the problems posed by the study of zetafunctions of shimura varieties. Introduction to the arithmetic theory of automorphic. Download product flyer download highresolution cover. Automorphic representations, shimura varieties, and. He was known for developing the theory of complex multiplication of abelian varieties and shimura varieties, as well as posing the. The intent of this book is to survey major interesting results and current trends in.
Introduction to the arithmetic theory of automorphic functions. The analytic theory of lfunctions has many applications to number theory and automorphic forms and visa versa. Periods and special values of lfunctions 3 strictly dividing n. 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. Automorphic forms, shimura varieties, galois representations and lfunctions msri workshop december 15, 2014 1 argos seminar on intersections of modular correspondences bonn, 20032004, ast erisque 312, soci et e. This intersection theory on the associated arithmetic surface was examined in detail by kudla, rapoport, and yang 14 and reduces in our case to computations of hilbert symbols.
This is not only the origin of those functions, but one of the most essential points to which we may ascribe the significance of elliptic modular functions in number theory. Goro shimura, shimura goro, 23 february 1930 3 may 2019 was a japanese mathematician and michael henry strater professor emeritus of mathematics at princeton university who worked in number theory, automorphic forms, and arithmetic geometry. In his introduction to this book, the author tells of how in 1980 he had the insight that the padic l functions attached to automorphic forms on various different algebraic groups should be functions of several variables. Number theory and its applicationsedited by cheon seoung ryoo. The geometry and cohomology of some simple shimura. 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. Also, let me remark that im told that the term automorphic vector bundle was invented in a conversation between michael harris and jim milne in the second half of the 80s. M the weierstrass function is really a function of two things. Purchase applications of functional analysis and operator theory, volume 200 2nd edition. We compute rings of automorphic forms on these curves. On certain lfunctions conference in honor of freydoon shahidi july 2327, 2007 purdue university. Automorphic functions and number theory goro shimura springer.
Automorphic representations, shimura varieties, and motives. Automorphic functions and number theory springerlink. This book systematically studies most of the numerous types of approximation properties that have been important in recent years. Lang, abelian varieties, interscience, new york, 1959.
In the course of the study of the zeta functions, many important results, which were obtained by the author during 19602000, are exposed. Automorphic forms and the cohomology of vector bundles on shimura varieties, michael harris. The analytic theory of theta functions and abelian varieties is systematically treated in 9 a. The journal of number theory jnt features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. Automorphic forms and cohomology theories on shimura curves of small discriminant michael hill, tyler lawson. Cohomology of arithmetic groups, lfunctions and automorphic forms, t. March 6, 2009 abstract we apply luries theorem to produce spectra associated to 1dimensional formal group laws on the shimura curves of discriminants 6, 10, and 14. For one thing, while in high school, i had acquired a decent amount of. Specifically to the ramanujan conjectures and other spectral problems associated with quotients of homogeneous spaces and to arithmetical problems such as quadratic forms. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. Citeseerx arithmeticity in the theory of automorphic forms. The school will be held at the fields institute in toronto and will be aimed at graduate students and mathematicians within five years of their ph.
Arithmeticity in the theory of automorphic forms, by goro. The fundamental material of abelian varieties is presented by 7 a. Let k f denote the sub eld of c generated by the a n. The geometry and cohomology of some simple shimura varieties.
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