Introduction to the arithmetic theory of automorphic functions. It turns out that the split exceptional group g 2, and certain forms of the other exceptional groups, possess a similar very special class. Secondly, we attach padic lfunctions to triples of ordinary padic. Automorphic functions and number theory book, 1968. Introduction to shimura varieties and automorphic vector bundles niccol o ronchetti february 2017 1 motivation lets recall rst the classical story.
Citeseerx arithmeticity in the theory of automorphic forms. Elliptic curves and the fundamental theorems of the classical theory of complex multiplication. Period relations for standard l functions of symplectic type. It had been my intention to survey the problems posed by the study of zetafunctions of shimura varieties. An introduction to the theory of automorphic functions. It is also beautifully structured and very wellwritten, if compactly. Introduction to arithmetic theory of automorphic functions.
Shimura varieties are not algebraic varieties but are families of algebraic varieties. One considers the upper half plane h and wants to study modular functions, or modular forms f. Automorphic forms and cohomology theories on shimura curves. Because of their brevity, many proofs have been omitted or only sketched. Automorphic forms and cohomology theories on shimura. Introduction to shimura varieties and automorphic vector bundles. This book introduces the reader to the subject and in particular to elliptic modular forms with emphasis on their number theoretical aspects. 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. Introduction to the arithmetic theory of automorphic. This book introduces the reader to the subject and in particular to elliptic modular forms with emphasis on their numbertheoretical aspects. Numerous and frequentlyupdated resource results are available from this search. Milne says that to show that the zeta function of a shimura variety is automorphic, you need 1 a description of the points of the variety over a. Triple product padic lfunctions for shimura curves over. The twentyfifth ams summer research institute was devoted to automorphic forms, representations and lfunctions.
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. Secondly, we attach padic l functions to triples of ordinary padic families of. Shimura varieties, galois representation and automorphic forms. Shimura,arithmeticity in the theory of automorphic forms. Using a recent geometric approach developed by andreatta and iovita we construct several variables padic families of finite slope quaternionic automorphic forms over f. Goro shimura, introduction to the arithmetic theory of automorphic functions larry joel goldstein. This institute emphasized representations so that, at least formally, the primary.
The very conception of a prime number goes back to antiquity, although it is not. These notes are an introduction to the theory of shimura varieties from the point of view of deligne and langlands. This would bea mammoth task, and limitations oftimeand energyhave considerably reducedthe compassofthis report. I dont quite see it though as an outsider, i may miss certain implications. An elementary construction of shimura varieties as moduli of abelian schemes. The second author would like to thank mark behrens, benjamin howard, niko naumann, and john voight for conversations related to this material. Automorphic functions and number theory ebook, 1968. The area of automorphic representations is a natural continuation of studies in the 19th and 20th centuries on number theory and modular forms. Berlin, heidelberg, new york, springerverlag, 1968. Goro shimura, introduction to the arithmetic theory of automorphic functions. Introduction to the arithmetic theory of automorphic functions shimura scan. The hodgetate period map is an important, new tool for studying the geometry of shimura varieties, padic automorphic forms and torsion classes in the cohomology of shimura varieties. B includes a as the 0dimensional special case of canonical models.
In addition, i was not aware of weils paper on the hecke theory or of the taniyama conjecture. It is well known that shimuras mathematics developed by stages. A guiding principle is a reciprocity law relating infinite dimensional automorphic representations with finite dimensional galois representations. The theory of multiple dirichlet series dirichlet series in several complex variables introduced in 1980s is now emerging as an important tool in obtaining sharp growth estimates for zeta and lfunctions, an important classical problem in number theory with applications to algebraic geometry. However, since this aim has been erected as one of the topmost paradigm of research in number theory, reasons for formulating motivations for studying automorphic. He was known for developing the theory of complex multiplication of abelian varieties and shimura.
For a nice treatment of the theory of automorphic functions box 2. Shimura curves are the onedimensional shimura varieties. 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. Find all the books, read about the author, and more. The connected components of these varieties are the canonical models of the preceding section. Automorphic forms and cohomology theories on shimura curves of small discriminant michael hilla. Automorphic representations, shimura varieties, and. Conference on eisenstein series and applications, american institute of mathematics, aug. For a prime p and a prime power q pk, we write f q for the. The coefficients of automorphic \l\functions attached to groups anisotropic over \\mathbb r\ can be interpreted in an elementary way as in a little bit of number theory.
On the zetafunctions of the algebraic curves uniformized by certain automorphic functions. Automorphic functions are also of importance in number theory. Automorphic functions and number theory goro shimura download bok. Shimura, goro, 1930automorphic functions and number theory. Automorphic functions and number theory springerlink. University of chicago number theory seminar, april 2016. 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. Nevertheless, it is not the fundamental lemma as such that is critical for the analytic theory of automorphic forms and for the arithmetic of shimura varieties. Shimuraon analytic families of polarized abelian varieties and automorphic functions, ann. Varieties number theory seminar, berkeley, fall 2016 sug woo shin the main goal is to understand scholzes paper sch15, concentrating on the cohomology of siegel modular varieties. He is the author of analy tic number theory prenticehall 1971, and abstract algebra, a first course prenticehall, to appear. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. Classical modular forms are very special automorphic functions for the group gl2, and similarly holomorphic siegel modular forms are very special automorphic functions for the group gsp2n.
To show that all lfunctions associated to shimura varieties thus to any motive defined by a shimura variety can be expressed in terms of the automorphic lfunctions of his paper of 1970 is weaker, even very much weaker, than to show that all motivic lfunctions are equal to such lfunctions. Automorphic functions on the upper half plane, especially modular functions. Automorphic forms and cohomology theories on shimura curves of small discriminant michael hill. Automorphic functions and number theory goro shimura. Berlin, heidelberg, new york, springerverlag, 1968 dlc 680252 ocolc443735. This vision encompasses the artin and shimurataniyama conjectures, both of which played a key role in wiles proof of fermat last theorem. Automorphic functions and number theory lecture notes in. 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. Introduction to shimura varieties and automorphic vector. Period relations for standard lfunctions of symplectic type. The history of the prime number theorem provides a beautiful example of the way in which great ideas develop and interrelate, feeding upon one another ultimately to yield a coherent theory which rather completely explains observed phenomena.
Automorphic functions and number theory lecture notes in mathematics 1968th edition. He was known for developing the theory of complex multiplication of abelian varieties and shimura varieties, as well as posing the. 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. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle. Further reading for shimuras approach, i suggest looking.
Automorphic representations, shimura varieties, and motives. 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. Over number fields, i t is mainly concerned with problems on shimura varieties. On the zeta functions of the algebraic curves uniformized by certain automorphic functions. C critical values of zeta functions and periods of automorphic forms. A main goal of the institute was the discussion of the lfunctions 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 lfunctions of algebraic number theory and algebraic geometry, such as artin l. Galois representation, shimura variety, special values of lfunctions. Selected work of dihua jiang some of the papers are available upon request on automorphic forms, shimura varieties, special lvalues, and arithmetic problems. At the time of writing they are the only tool in the study of zetafunctions of algebraic varieties 11.
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. Notations and conventions unless indicated otherwise, vector spaces are assumed to be. This is an introduction to the theory of shimura varieties, or, in other words, to the arithmetic theory of automorphic functions and holomorphic automorphic forms. B the theory of canonical models shimura varieties.