eBook ISBN:  9780821876749 
Product Code:  CONM/86.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
eBook ISBN:  9780821876749 
Product Code:  CONM/86.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 

Book DetailsContemporary MathematicsVolume: 86; 1989; 266 ppMSC: Primary 11; Secondary 20; 22
The Langlands Program summarizes those parts of mathematical research belonging to the representation theory of reductive groups and to class field theory. These two topics are connected by the vision that, roughly speaking, the irreducible representations of the general linear group may well serve as parameters for the description of all number fields. In the local case, the base field is a given \(p\)adic field \(K\) and the extension theory of \(K\) is seen as determined by the irreducible representations of the absolute Galois group \(G_K\) of \(K\). Great progress has been made in establishing correspondence between the supercuspidal representations of \(GL(n,K)\) and those irreducible representations of \(G_K\) whose degrees divide \(n\). Despite these advances, no book or paper has presented the different methods used or even collected known results.
This volume contains the proceedings of the conference “Representation Theory and Number Theory in Connection with the Local Langlands Conjecture,” held in December 1985 at the University of Augsburg. The program of the conference was divided into two parts: (i) the representation theory of local division algebras and local Galois groups, and the Langlands conjecture in the tame case; and (ii) new results, such as the case \(n=p\), the matching theorem, principal orders, tame Deligne representations, classification of representations of \(GL(n)\), and the numerical Langlands conjecture. The collection of papers in this volume provides an excellent account of the current state of the local Langlands Program.

Table of Contents

Articles

E. Becker and B. Külshammer — The irreducible representations of the multiplicative group of a tame division algebra over a local field (following H. Koch and E.W. Zink) [ MR 987011 ]

J. Rohlfs — Sequences of Eisenstein polynomials and arithmetic in local division algebras [ MR 987012 ]

Moshe Jarden — Koch’s classification of the primitive representations of a Galois group of a local field [ MR 987013 ]

M. Lorenz — On the numerical local Langlands conjecture [ MR 987014 ]

H. Opolka — Ramifications of Weilrepresentations of local Galois groups [ MR 987015 ]

W. Willems — Representations of certain group extensions [ MR 987016 ]

J. Brinkhuis — Trace calculations [ MR 987017 ]

G. R. Everest — Root numbers—the tame case [ MR 987018 ]

K. Wingberg — Representations of locally profinite groups [ MR 987019 ]

U. Jannsen — The Theorems of Bernštein and Zelevinskii

S. M. J. Wilson — Principal Orders and Congruence Gauss Sums

J. Queyrut — The Functional Equation $\epsilon $Factors

M. Taylor — Root Numbers and the Local Langlands Conjecture

Phil Kutzko — On the Exceptional Representations of $\mathrm {GL}_n$

L. Corwin — Characters of Representations of $D^*_n$ (Tamely Ramified Case)

P. J. Sally, Jr. — Matching and Formal Degrees for Division Algebras and $\mathrm {GL}_n$ over a $p$adic Field

A. Fröhlich — Tame Representations and Base Change

C. J. Bushnell — Gauss Sums and Supercuspidal Representations of $\mathrm {GL}_n$

P. Gérardin and WenCh’ing Winnie Li — Identities on Degree Two Gamma Factors

A. Moy — A Conjecture on Minimal Ktypes for $\mathrm {GL}_n$ over a $p$adic Field

G. Henniart — Preuve de la Conjecture de Langlands Locale Numerique pour $\mathrm {GL}(n)$


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The Langlands Program summarizes those parts of mathematical research belonging to the representation theory of reductive groups and to class field theory. These two topics are connected by the vision that, roughly speaking, the irreducible representations of the general linear group may well serve as parameters for the description of all number fields. In the local case, the base field is a given \(p\)adic field \(K\) and the extension theory of \(K\) is seen as determined by the irreducible representations of the absolute Galois group \(G_K\) of \(K\). Great progress has been made in establishing correspondence between the supercuspidal representations of \(GL(n,K)\) and those irreducible representations of \(G_K\) whose degrees divide \(n\). Despite these advances, no book or paper has presented the different methods used or even collected known results.
This volume contains the proceedings of the conference “Representation Theory and Number Theory in Connection with the Local Langlands Conjecture,” held in December 1985 at the University of Augsburg. The program of the conference was divided into two parts: (i) the representation theory of local division algebras and local Galois groups, and the Langlands conjecture in the tame case; and (ii) new results, such as the case \(n=p\), the matching theorem, principal orders, tame Deligne representations, classification of representations of \(GL(n)\), and the numerical Langlands conjecture. The collection of papers in this volume provides an excellent account of the current state of the local Langlands Program.

Articles

E. Becker and B. Külshammer — The irreducible representations of the multiplicative group of a tame division algebra over a local field (following H. Koch and E.W. Zink) [ MR 987011 ]

J. Rohlfs — Sequences of Eisenstein polynomials and arithmetic in local division algebras [ MR 987012 ]

Moshe Jarden — Koch’s classification of the primitive representations of a Galois group of a local field [ MR 987013 ]

M. Lorenz — On the numerical local Langlands conjecture [ MR 987014 ]

H. Opolka — Ramifications of Weilrepresentations of local Galois groups [ MR 987015 ]

W. Willems — Representations of certain group extensions [ MR 987016 ]

J. Brinkhuis — Trace calculations [ MR 987017 ]

G. R. Everest — Root numbers—the tame case [ MR 987018 ]

K. Wingberg — Representations of locally profinite groups [ MR 987019 ]

U. Jannsen — The Theorems of Bernštein and Zelevinskii

S. M. J. Wilson — Principal Orders and Congruence Gauss Sums

J. Queyrut — The Functional Equation $\epsilon $Factors

M. Taylor — Root Numbers and the Local Langlands Conjecture

Phil Kutzko — On the Exceptional Representations of $\mathrm {GL}_n$

L. Corwin — Characters of Representations of $D^*_n$ (Tamely Ramified Case)

P. J. Sally, Jr. — Matching and Formal Degrees for Division Algebras and $\mathrm {GL}_n$ over a $p$adic Field

A. Fröhlich — Tame Representations and Base Change

C. J. Bushnell — Gauss Sums and Supercuspidal Representations of $\mathrm {GL}_n$

P. Gérardin and WenCh’ing Winnie Li — Identities on Degree Two Gamma Factors

A. Moy — A Conjecture on Minimal Ktypes for $\mathrm {GL}_n$ over a $p$adic Field

G. Henniart — Preuve de la Conjecture de Langlands Locale Numerique pour $\mathrm {GL}(n)$