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An Introduction to the Langlands Program Paperback – May 20, 2003

ISBN-13: 978-0817632113 ISBN-10: 0817632115 Edition: 2004th

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An Introduction to the Langlands Program + Automorphic Forms and Representations (Cambridge Studies in Advanced Mathematics) + Automorphic Forms on Adele Groups. (AM-83) (Annals of Mathematics Studies)
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Product Details

  • Paperback: 288 pages
  • Publisher: Birkhäuser; 2004 edition (May 20, 2003)
  • Language: English
  • ISBN-10: 0817632115
  • ISBN-13: 978-0817632113
  • Product Dimensions: 6.1 x 0.7 x 9.2 inches
  • Shipping Weight: 14.1 ounces (View shipping rates and policies)
  • Average Customer Review: 4.0 out of 5 stars  See all reviews (1 customer review)
  • Amazon Best Sellers Rank: #1,210,769 in Books (See Top 100 in Books)

Editorial Reviews


From the reviews:

"The six chapters of this monograph give 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. First-year graduate students and researchers will benefit from this beautiful text."

--Zentralblatt Math

". . . the present volume constitutes the most readable entree into the subject to date, suitable both for serious reading and for browsing, and should attract a new generation to this exciting subject. . . . Recommended."


“I suspect this book will find its way into the hands of many graduate students. Perhaps it will also motivate a few of them to learn more, get involved, and make their own contributions.” (MAA REVIEWS)

From the Back Cover

For the past several decades the theory of automorphic forms has become a major focal point of development in number theory and algebraic geometry, with applications in many diverse areas, including combinatorics and mathematical physics.

The twelve chapters of this monograph present 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.

Key features of this self-contained presentation:

       A variety of areas in number theory from the classical zeta function up to the Langlands program are covered.

       The exposition is systematic, with each chapter focusing on a particular topic devoted to special cases of the program:

• Basic zeta function of Riemann and its generalizations to Dirichlet and Hecke L-functions, class field theory and some topics on classical automorphic functions (E. Kowalski)

• A study of the conjectures of Artin and Shimura–Taniyama–Weil (E. de Shalit)

• An examination of classical modular (automorphic) L-functions as GL(2) functions,   bringing into play the theory of representations (S.S. Kudla)

• Selberg's theory of the trace formula, which is a way to study automorphic representations (D. Bump)

• Discussion of cuspidal automorphic representations of GL(2,(A)) leads to Langlands theory for GL(n) and the importance of the Langlands dual group (J.W. Cogdell)

• An introduction to the geometric Langlands program, a new and active area of research that permits using powerful methods of algebraic geometry to construct automorphic sheaves (D. Gaitsgory)

Graduate students and researchers will benefit from this beautiful text.

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Most Helpful Customer Reviews

8 of 8 people found the following review helpful By Dr. Lee D. Carlson HALL OF FAMEVINE VOICE on March 14, 2012
Format: Paperback Verified Purchase
Very loosely speaking, the Langlands program can be viewed as an attempt to study to what extent L-functions are really different from each other. L-functions have their origins in number theory as a method for counting or manipulating various arithmetic objects, such as prime numbers. As such the use of L-functions is somewhat surprising, since they are objects coming from analysis, and not number theory. The connection of L-functions to arithmetic is however made more believable if one remembers how the gamma function is defined for all real numbers instead of just the natural numbers. Once this definition is made the gamma function satisfies interesting recursive relations, with similar types of relations being satisfied by more "general" L-functions.

The most famous example of an L-function is of course the Riemann zeta function, which was used by the mathematician Leonhard Euler to give an "analytic" proof that there are an infinite number of primes. Other fascinating properties of this L-function are shown in the first article of this book, in particular its analytic continuation in the complex plane and the functional equation that it satisfies. The author views the presence of the gamma function in this functional equation as being somewhat mysterious, and he points to another article in the book as serving as the best resolution of this mystery. This article concerns the thesis of the mathematician John Tate, and is advertised as a way of unifying the analytical continuation and functional equation of L-functions.
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