**Author**: Colin J. Bushnell

**Publisher:**Springer Science & Business Media

**ISBN:**354031511X

**Category :**Mathematics

**Languages :**en

**Pages :**340

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## The Local Langlands Conjecture for GL(2)

**Author**: Colin J. Bushnell

**Publisher:** Springer Science & Business Media

**ISBN:** 354031511X

**Category : **Mathematics

**Languages : **en

**Pages : **340

**Book Description**

The Local Langlands Conjecture for GL(2) contributes an unprecedented text to the so-called Langlands theory. It is an ambitious research program of already 40 years and gives a complete and self-contained proof of the Langlands conjecture in the case n=2. It is aimed at graduate students and at researchers in related fields. It presupposes no special knowledge beyond the beginnings of the representation theory of finite groups and the structure theory of local fields.

## The Local Langlands Conjecture for GL(2)

**Author**: Colin J. Bushnell

**Publisher:** Springer Science & Business Media

**ISBN:** 354031511X

**Category : **Mathematics

**Languages : **en

**Pages : **340

**Book Description**

The Local Langlands Conjecture for GL(2) contributes an unprecedented text to the so-called Langlands theory. It is an ambitious research program of already 40 years and gives a complete and self-contained proof of the Langlands conjecture in the case n=2. It is aimed at graduate students and at researchers in related fields. It presupposes no special knowledge beyond the beginnings of the representation theory of finite groups and the structure theory of local fields.

## Representation Theory and Number Theory in Connection with the Local Langlands Conjecture

**Author**: Jürgen Ritter

**Publisher:** American Mathematical Soc.

**ISBN:** 0821850938

**Category : **Mathematics

**Languages : **en

**Pages : **266

**Book Description**

## Automorphic Forms, Representations and $L$-Functions

**Author**: Armand Borel

**Publisher:** American Mathematical Soc.

**ISBN:** 0821814370

**Category : **Mathematics

**Languages : **en

**Pages : **382

**Book Description**

Part 2 contains sections on Automorphic representations and $L$-functions, Arithmetical algebraic geometry and $L$-functions

## To an Effective Local Langlands Correspondence

**Author**: Colin J. Bushnell

**Publisher:** American Mathematical Soc.

**ISBN:** 082189417X

**Category : **Mathematics

**Languages : **en

**Pages : **88

**Book Description**

Let F be a non-Archimedean local field. Let \mathcal{W}_{F} be the Weil group of F and \mathcal{P}_{F} the wild inertia subgroup of \mathcal{W}_{F}. Let \widehat {\mathcal{W}}_{F} be the set of equivalence classes of irreducible smooth representations of \mathcal{W}_{F}. Let \mathcal{A}^{0}_{n}(F) denote the set of equivalence classes of irreducible cuspidal representations of \mathrm{GL}_{n}(F) and set \widehat {\mathrm{GL}}_{F} = \bigcup _{n\ge 1} \mathcal{A}^{0}_{n}(F). If \sigma \in \widehat {\mathcal{W}}_{F}, let ^{L}{\sigma }\in \widehat {\mathrm{GL}}_{F} be the cuspidal representation matched with \sigma by the Langlands Correspondence. If \sigma is totally wildly ramified, in that its restriction to \mathcal{P}_{F} is irreducible, the authors treat ^{L}{\sigma} as known. From that starting point, the authors construct an explicit bijection \mathbb{N}:\widehat {\mathcal{W}}_{F} \to \widehat {\mathrm{GL}}_{F}, sending \sigma to ^{N}{\sigma}. The authors compare this "naïve correspondence" with the Langlands correspondence and so achieve an effective description of the latter, modulo the totally wildly ramified case. A key tool is a novel operation of "internal twisting" of a suitable representation \pi (of \mathcal{W}_{F} or \mathrm{GL}_{n}(F)) by tame characters of a tamely ramified field extension of F, canonically associated to \pi. The authors show this operation is preserved by the Langlands correspondence.

## Canadian Journal of Mathematics

**Author**:

**Publisher:**

**ISBN:**

**Category : **

**Languages : **en

**Pages : **224

**Book Description**

## An Introduction to the Langlands Program

**Author**: Joseph Bernstein

**Publisher:** Springer Science & Business Media

**ISBN:** 0817682260

**Category : **Mathematics

**Languages : **en

**Pages : **281

**Book Description**

This book presents a broad, user-friendly introduction to the Langlands program, that is, the theory of automorphic forms and its connection with the theory of L-functions and other fields of mathematics. Each of the twelve chapters focuses on a particular topic devoted to special cases of the program. The book is suitable for graduate students and researchers.

## Advances in the Theory of Numbers

**Author**: Ayşe Alaca

**Publisher:** Springer

**ISBN:** 1493932012

**Category : **Mathematics

**Languages : **en

**Pages : **235

**Book Description**

The theory of numbers continues to occupy a central place in modern mathematics because of both its long history over many centuries as well as its many diverse applications to other fields such as discrete mathematics, cryptography, and coding theory. The proof by Andrew Wiles (with Richard Taylor) of Fermat’s last theorem published in 1995 illustrates the high level of difficulty of problems encountered in number-theoretic research as well as the usefulness of the new ideas arising from its proof. The thirteenth conference of the Canadian Number Theory Association was held at Carleton University, Ottawa, Ontario, Canada from June 16 to 20, 2014. Ninety-nine talks were presented at the conference on the theme of advances in the theory of numbers. Topics of the talks reflected the diversity of current trends and activities in modern number theory. These topics included modular forms, hypergeometric functions, elliptic curves, distribution of prime numbers, diophantine equations, L-functions, Diophantine approximation, and many more. This volume contains some of the papers presented at the conference. All papers were refereed. The high quality of the articles and their contribution to current research directions make this volume a must for any mathematics library and is particularly relevant to researchers and graduate students with an interest in number theory. The editors hope that this volume will serve as both a resource and an inspiration to future generations of researchers in the theory of numbers.

## Eisenstein Series and Automorphic L-functions

**Author**: Freydoon Shahidi

**Publisher:** American Mathematical Soc.

**ISBN:** 0821849891

**Category : **Mathematics

**Languages : **en

**Pages : **210

**Book Description**

This book presents a treatment of the theory of $L$-functions developed via the theory of Eisenstein series and their Fourier coefficients. The author is a co-developer of the important Langlands-Shahidi method. This account of the theory is ideal for graduate students and researchers interested in the Langlands program in automorphic forms and its connections with number theory.

## Modular Forms and Fermat’s Last Theorem

**Author**: Gary Cornell

**Publisher:** Springer Science & Business Media

**ISBN:** 9780387946092

**Category : **Mathematics

**Languages : **en

**Pages : **608

**Book Description**

A collection of expanded versions of lectures given at an instructional conference on number theory and arithmetic geometry held at Boston University. The purpose of the conference, and indeed this book, is to introduce and explain the many ideas and techniques used by Wiles in his proof, and to explain how his result can be combined with Ribet's theorem and ideas of Frey and Serre to show, at long last, that Fermat's Last Theorem is true. The book begins with an overview of the complete proof, theory of elliptic curves, modular functions, modular curves, Galois cohomology, and finite group schemes. In recognition of the historical significance of Fermat's Last Theorem, the volume concludes by reflecting on the history of the problem, while placing Wiles' theorem into a more general Diophantine context suggesting future applications.

## Derived Langlands: Monomial Resolutions Of Admissible Representations

**Author**: Victor P Snaith

**Publisher:** World Scientific

**ISBN:** 9813275766

**Category : **Mathematics

**Languages : **en

**Pages : **356

**Book Description**

The Langlands Programme is one of the most important areas in modern pure mathematics. The importance of this volume lies in its potential to recast many aspects of the programme in an entirely new context. For example, the morphisms in the monomial category of a locally p-adic Lie group have a distributional description, due to Bruhat in his thesis. Admissible representations in the programme are often treated via convolution algebras of distributions and representations of Hecke algebras. The monomial embedding, introduced in this book, elegantly fits together these two uses of distribution theory. The author follows up this application by giving the monomial category treatment of the Bernstein Centre, classified by Deligne-Bernstein-Zelevinsky.This book gives a new categorical setting in which to approach well-known topics. Therefore, the context used to explain examples is often the more generally accessible case of representations of finite general linear groups. For example, Galois base-change and epsilon factors for locally p-adic Lie groups are illustrated by the analogous Shintani descent and Kondo-Gauss sums, respectively. General linear groups of local fields are emphasized. However, since the philosophy of this book is essentially that of homotopy theory and algebraic topology, it includes a short appendix showing how the buildings of Bruhat-Tits, sufficient for the general linear group, may be generalised to the tom Dieck spaces (now known as the Baum-Connes spaces) when G is a locally p-adic Lie group.The purpose of this monograph is to describe a functorial embedding of the category of admissible k-representations of a locally profinite topological group G into the derived category of the additive category of the admissible k-monomial module category. Experts in the Langlands Programme may be interested to learn that when G is a locally p-adic Lie group, the monomial category is closely related to the category of topological modules over a sort of enlarged Hecke algebra with generators corresponding to characters on compact open modulo the centre subgroups of G. Having set up this functorial embedding, how the ingredients of the celebrated Langlands Programme adapt to the context of the derived monomial module category is examined. These include automorphic representations, epsilon factors and L-functions, modular forms, Weil-Deligne representations, Galois base change and Hecke operators.

Book PDF Download Digital Journal

The Local Langlands Conjecture for GL(2) contributes an unprecedented text to the so-called Langlands theory. It is an ambitious research program of already 40 years and gives a complete and self-contained proof of the Langlands conjecture in the case n=2. It is aimed at graduate students and at researchers in related fields. It presupposes no special knowledge beyond the beginnings of the representation theory of finite groups and the structure theory of local fields.

The Local Langlands Conjecture for GL(2) contributes an unprecedented text to the so-called Langlands theory. It is an ambitious research program of already 40 years and gives a complete and self-contained proof of the Langlands conjecture in the case n=2. It is aimed at graduate students and at researchers in related fields. It presupposes no special knowledge beyond the beginnings of the representation theory of finite groups and the structure theory of local fields.

Part 2 contains sections on Automorphic representations and $L$-functions, Arithmetical algebraic geometry and $L$-functions

Let F be a non-Archimedean local field. Let \mathcal{W}_{F} be the Weil group of F and \mathcal{P}_{F} the wild inertia subgroup of \mathcal{W}_{F}. Let \widehat {\mathcal{W}}_{F} be the set of equivalence classes of irreducible smooth representations of \mathcal{W}_{F}. Let \mathcal{A}^{0}_{n}(F) denote the set of equivalence classes of irreducible cuspidal representations of \mathrm{GL}_{n}(F) and set \widehat {\mathrm{GL}}_{F} = \bigcup _{n\ge 1} \mathcal{A}^{0}_{n}(F). If \sigma \in \widehat {\mathcal{W}}_{F}, let ^{L}{\sigma }\in \widehat {\mathrm{GL}}_{F} be the cuspidal representation matched with \sigma by the Langlands Correspondence. If \sigma is totally wildly ramified, in that its restriction to \mathcal{P}_{F} is irreducible, the authors treat ^{L}{\sigma} as known. From that starting point, the authors construct an explicit bijection \mathbb{N}:\widehat {\mathcal{W}}_{F} \to \widehat {\mathrm{GL}}_{F}, sending \sigma to ^{N}{\sigma}. The authors compare this "naïve correspondence" with the Langlands correspondence and so achieve an effective description of the latter, modulo the totally wildly ramified case. A key tool is a novel operation of "internal twisting" of a suitable representation \pi (of \mathcal{W}_{F} or \mathrm{GL}_{n}(F)) by tame characters of a tamely ramified field extension of F, canonically associated to \pi. The authors show this operation is preserved by the Langlands correspondence.

This book presents a broad, user-friendly introduction to the Langlands program, that is, the theory of automorphic forms and its connection with the theory of L-functions and other fields of mathematics. Each of the twelve chapters focuses on a particular topic devoted to special cases of the program. The book is suitable for graduate students and researchers.

The theory of numbers continues to occupy a central place in modern mathematics because of both its long history over many centuries as well as its many diverse applications to other fields such as discrete mathematics, cryptography, and coding theory. The proof by Andrew Wiles (with Richard Taylor) of Fermat’s last theorem published in 1995 illustrates the high level of difficulty of problems encountered in number-theoretic research as well as the usefulness of the new ideas arising from its proof. The thirteenth conference of the Canadian Number Theory Association was held at Carleton University, Ottawa, Ontario, Canada from June 16 to 20, 2014. Ninety-nine talks were presented at the conference on the theme of advances in the theory of numbers. Topics of the talks reflected the diversity of current trends and activities in modern number theory. These topics included modular forms, hypergeometric functions, elliptic curves, distribution of prime numbers, diophantine equations, L-functions, Diophantine approximation, and many more. This volume contains some of the papers presented at the conference. All papers were refereed. The high quality of the articles and their contribution to current research directions make this volume a must for any mathematics library and is particularly relevant to researchers and graduate students with an interest in number theory. The editors hope that this volume will serve as both a resource and an inspiration to future generations of researchers in the theory of numbers.

This book presents a treatment of the theory of $L$-functions developed via the theory of Eisenstein series and their Fourier coefficients. The author is a co-developer of the important Langlands-Shahidi method. This account of the theory is ideal for graduate students and researchers interested in the Langlands program in automorphic forms and its connections with number theory.

A collection of expanded versions of lectures given at an instructional conference on number theory and arithmetic geometry held at Boston University. The purpose of the conference, and indeed this book, is to introduce and explain the many ideas and techniques used by Wiles in his proof, and to explain how his result can be combined with Ribet's theorem and ideas of Frey and Serre to show, at long last, that Fermat's Last Theorem is true. The book begins with an overview of the complete proof, theory of elliptic curves, modular functions, modular curves, Galois cohomology, and finite group schemes. In recognition of the historical significance of Fermat's Last Theorem, the volume concludes by reflecting on the history of the problem, while placing Wiles' theorem into a more general Diophantine context suggesting future applications.

The Langlands Programme is one of the most important areas in modern pure mathematics. The importance of this volume lies in its potential to recast many aspects of the programme in an entirely new context. For example, the morphisms in the monomial category of a locally p-adic Lie group have a distributional description, due to Bruhat in his thesis. Admissible representations in the programme are often treated via convolution algebras of distributions and representations of Hecke algebras. The monomial embedding, introduced in this book, elegantly fits together these two uses of distribution theory. The author follows up this application by giving the monomial category treatment of the Bernstein Centre, classified by Deligne-Bernstein-Zelevinsky.This book gives a new categorical setting in which to approach well-known topics. Therefore, the context used to explain examples is often the more generally accessible case of representations of finite general linear groups. For example, Galois base-change and epsilon factors for locally p-adic Lie groups are illustrated by the analogous Shintani descent and Kondo-Gauss sums, respectively. General linear groups of local fields are emphasized. However, since the philosophy of this book is essentially that of homotopy theory and algebraic topology, it includes a short appendix showing how the buildings of Bruhat-Tits, sufficient for the general linear group, may be generalised to the tom Dieck spaces (now known as the Baum-Connes spaces) when G is a locally p-adic Lie group.The purpose of this monograph is to describe a functorial embedding of the category of admissible k-representations of a locally profinite topological group G into the derived category of the additive category of the admissible k-monomial module category. Experts in the Langlands Programme may be interested to learn that when G is a locally p-adic Lie group, the monomial category is closely related to the category of topological modules over a sort of enlarged Hecke algebra with generators corresponding to characters on compact open modulo the centre subgroups of G. Having set up this functorial embedding, how the ingredients of the celebrated Langlands Programme adapt to the context of the derived monomial module category is examined. These include automorphic representations, epsilon factors and L-functions, modular forms, Weil-Deligne representations, Galois base change and Hecke operators.