On automaorphic functions and the reciprocity law in a number field. by Tomio Kubota

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Published by Kinojuniya in [Tokyo] .

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SeriesLectures in mathematics - Kyoto University, Dept. of Mathematics -- v. 2
ID Numbers
Open LibraryOL14805280M

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Get this from a library. On automorphic functions: and the reciprocity law in a number field. [T Kubota]. On automorphic functions and the reciprocity law in a. On automorphic functions and the reciprocity law in a number field. By Tomio Kubota. Download PDF (15 MB) Introduction§1.

GL(2) over a totally imaginary field.§2. Hecke operators and automorphic functions.§3. Construction of a covering group.§4. Hecke ring of the metaplectic group.§5. Eisenstein series.§6.

Unitary representations Author: Tomio Kubota. “On Automorphic Functions and the Reciprocity Law in A Number Field,” Tokyo Kinokumiya Book Store Ltd., ().

Google Scholar [P] Patterson, S.J. “On Dirichlet Series Associated with Cubic Gauss Sums,” J. Reine Angew. Math. –, (). Google ScholarCited by: After preliminaries--including a section, "Notation and Terminology"--the first part of On automaorphic functions and the reciprocity law in a number field.

book book deals with automorphic forms on such groups. In particular, their rationality over a number field is defined and discussed in connection with the group action; also the reciprocity law for the values of automorphic functions at CM-points is proved.

The theta function proof is also discussed in Dym and McKean's book "Fourier Series and Integrals" and in Richard Bellman's book "A Brief Introduction to Theta Functions." Bellman points out that theta reciprocity is a remarkable consequence of the fact that when the theta function is extended to two variables, both sides of the.

The Artin reciprocity law, which was established by Emil Artin in a series of papers (; ; ), is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and.

[KubotaBook] T. Kubota, On Automorphic Functions and the Reciprocity Law in a Number Field, Tokyo: Kinokuniya Book-Store Co. Ltd.,vol. Show bibtex @book {KubotaBook, MRKEY =. On automorphic functions and the reciprocity law in a number field.

Lectures in Mathematics, Department of Mathematics, Kyoto University, No. Kinokuniya Book-Store Co. Ltd., Tokyo, Google Scholar. classification of central simple algebras over number fields. HECKE (–). Introduced Hecke L-series generalizing both Dirichlet’s L-series and Dedekind’s zeta functions. ARTIN (–).

He found the “Artin reciprocity law”, which is the main theorem of class field theory (improvement of Takagi’s results). Automorphic Representations and L-Functions for the General Linear Group Volume 1 Published on: Author: giby Comment: 0 Automorphic Representations and L-Functions for the General.

The Artin reciprocity law applies to a Galois extension of an algebraic number field whose Galois group is abelian; it assigns L-functions to the one-dimensional representations of this Galois group, and states that these L-functions are identical to certain Dirichlet L-series or more general series (that is, certain analogues of the Riemann.

This book introduces the reader to the subject and in particular to elliptic modular forms with emphasis on their number-theoretical aspects. After two chapters geared toward elementary levels, there follows a detailed treatment of the theory of Hecke operators, which associate zeta functions to modular forms.

Reciprocity laws and density theorems, Richard Taylor, Shaw lecture (for a general audience) () 19pp. (Classical) Automorphic forms course page, Holger Then () Automorphic forms and Langlands program, Yannan Qiu, course notes taken by Robert Rhoades () pp. An elementary introduction to the local trace formulas of J.

Arthur. I The nonabelian reciprocity law is known in the function field case thanks to Drinfeld in the two-dimensional case, and Lafforgue in general.

I In the number field case, it is more difficult even to state the non abelian reciprocity law because there are automorphic forms that do. After preliminaries—including a section, “Notation and Terminology”—the first part of the book deals with automorphic forms on such groups.

In particular, their rationality over a number field is defined and discussed in connection with the group action; also the reciprocity law for the values of automorphic functions at CM-points is proved.

He found the first complete proofs of the quadratic reciprocity law. He studied the Gaussian integers Z„i“in order to find a quartic reciprocity law. He studied the classification of binary quadratic forms over Z, which is closely related to the problem of finding the class numbers of quadratic fields.

DIRICHLET (–). The book features extensive foundational material on the representation theory of GL(1) and GL(2) over local fields, the theory of automorphic representations, L-functions and advanced topics such as the Langlands conjectures, the Weil representation, the Rankin-Selberg method and the triple L-function, and examines this subject matter from.

Idea. Class field theory studies finite-dimensional abelian field extensions of number fields and of function fields, hence of global fields by relating them to the idele class group. Class field theory clarifies the origin of various reciprocity laws in number basic (one dimensional) class field theory stems from the ideas of Kronecker and Weber, and results of Hilbert soon after.

The seminar will meet on Fridays am to noon via Zoom according to the schedule below. Sep 25 Dennis Gaitsgory (Harvard) The stack of local systems with restricted variation and the passage from geometric to classical Langlands theory.

Arithmeticity in the Theory of Automorphic Forms About this Title. Goro Shimura, Princeton University, Princeton, NJ. Publication: Mathematical Surveys and Monographs Publication Year Volume 82 ISBNs: (print); (online). So I'm looking to see why the statement of class field theory that I know is essentially the same as a certain statement about L-functions, representations, or automorphic forms, in such a way that a more advanced mathematician could easily recognize the latter statement as Langlands in dimension 1.

2 Arthur & Gelbart - Lectures on automorphic L-functions: Part I in which the reader can find further information. More detailed discussion is given in various parts of the Corvallis Proceedings and in many of the other references we have cited.

PART I 1 STANDARD L-FUNCTIONS FOR GLn Let F be a fixed number field. Idea. Under the function field analogy it makes sense to regard a number field as the rational functions on an “arithmetic curve over F1”. Accordingly, there is a sensible version of the concept of genus of a curve for number fields.

This genus of a number field was originally introduced in in a maybe somewhat ad hoc is derived as being that definition which makes the Riemann.

Kubota, On automorphic functions and the reciprocity law in a number field, Lectures in Mathematics, Department of Mathematics, Kyoto University, No.

2 Kinokuniya Book. Algebraic cuspidal automorphic representations of unitary group appears in -adic cohomology of Shimura variety. In [35], Kottwitz proved a formula for the number of points on certain type of.

Abstract. Building on the topological foundations constructed in Part I, we now go on to address the homological algebra preparatory to the projected final arithmetical phase of our attack on the analytic proof of general reciprocity for a number field.

10 Geometrie Ramanujan Conjecture and Drinfeld Reciprocity Law* YUVAL Z. FLICKER AND DAVID A. KAZHDAN The purpose of this article is to describe and explain some of our recent work, which concerns, in particular, the following themes: The Ramanujan or purity conjecture for cuspidal automorphic forms with a super cuspidal component of GL(r) over a global field F of characteristic p.

Whittaker-Shintani functions for orthogonal groups Kato, Shin-ichi, Murase, Atsushi, and Sugano, Takashi, Tohoku Mathematical Journal, ; On an analogue of the Ichino–Ikeda conjecture for Whittaker coefficients on the metaplectic group Lapid, Erez and Mao, Zhengyu, Algebra & Number Theory, ; Metaplectic covers of Kac–Moody groups and Whittaker functions Patnaik, Manish M.

Author's comments: This note Funktorialität in der Theorie der automorphen Formen: Ihre Entdeckung und ihre Ziele was written as commentary to accompany the original letter in a collection of documents on reciprocity laws and algebraic number theory, to appear shortly. There is a curious ambiguity in the fifth section regarding the location of my office in the old Fine Hall and of a small.

Goro Shimura’s monograph, Introduction to the Arithmetic Theory of Automorphic Functions, published originally by Iwanami Shoten together with Princeton University Press, and now re-issued in paperback by Princeton, is one of the most important books in the is also beautifully structured and very well-written, if compactly.

It is unimaginable that a number theorist, be he a. 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: 1.

An elementary construction of Shimura varieties as moduli of abelian schemes. p-adic deformation theory of automorphic forms on Shimura varieties.

Then the reciprocity-law in the maximal abelian extension of an imaginary quadratic field is given as a certain commutativity of the action of the adeles with the specialization of the functions of $\scr F$. It gives a new point of view on this classical subject. The Artin reciprocity law, established by Emil Artin in a series of papers (; ; ), is a general theorem in number theory that forms a central part of the global class field theory.

[1] The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and. Publisher Summary. This chapter describes the generalization on the base change problem. The problem is connected with the class-field theory.

A generalization of the construction of I and B to arbitrary extensions would constitute a non-abelian class field theory; it would imply that the Dedekind zeta function of a given number field of a certain degree over another field is the L-function.

Editorial Reviews. From the reviews: "Hida views the study of the geometric Galois group of the Shimura tower, as a geometric reciprocity law. general goal of the book is to incorporate Shimura’s reciprocity law in a broader scheme of integral reciprocity laws which includes Iwasawa theory in its scope.

a beautiful and very useful reference for anybody interested in the Price: $ Presenting multidisciplinary methods (localization, Borcherds products, theory of special functions, Cremona maps, etc) for treating a range of partition functions, the book is intended for graduate students and young postdocs interested in the interaction between quantum field theory and mathematics related to automorphic forms, representation.

After preliminaries--including a section, ``Notation and Terminology''--the first part of the book deals with automorphic forms on such groups. In particular, their rationality over a number field. Automorphic Forms and L-Functions for the Group GL(n, R Of l-functions, by the approach known as “integral representations”, which goes back to hecke in the case of sl(2,r).

In this approach, the l-function l(s)in question is expressed as an integral of an automorphic form against a suitable integral kernel, depending on the complex. Let K be the Galois extension over the rational number field Q generated by and. Then its Galois group over Q is the dihedral group D 4 of order 8 and has the unique two-dimensional irreducible complex representation ψ.

In view of the theory of Hecke-Weil-Langlands, we know that ψ defines a cusp form of weight one (cf. Serre [6]). Abelian class field theory generalizes quadratic reciprocity laws for general number fields with abelian Galois groups, which connects class groups and Galois groups via Artin's reciprocity map.

that Langlands program is the non-abelian class field theory in the way of giving a criterions for splitting primes in a number field with non.that every ray class field over M can be generated, over a ray class field over F, by the special values of automorphic functions with respect to a congruence subgroup of P(o) at a regular fixed point of M (Main Theorem I ()).

An explicit reciprocity law for these class fields will be given as Main Theorem II. In the early years of the s, while I was visiting the Institute for Ad vanced Study (lAS) at Princeton as a postdoctoral member, I got a fascinating view, studying congruence modulo a prime among elliptic modular forms, that an automorphic L-function of a given algebraic group G should have a canon ical p-adic counterpart of several variables.

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