Lectures on the Arthur-Selberg Trace Formula

Overview

The Arthur-Selberg trace formula is an equality between two kinds of traces: the geometric terms given by the conjugacy classes of a group and the spectral terms given by the induced representations. In general, these terms require a truncation in order to converge, which leads to an equality of truncated kernels. The formulas are difficult in general and even the case of $GL$(2) is nontrivial. The book gives proof of Arthur's trace formula of the 1970s and 1980s, with special attention given to $GL$(2). The ...

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Brand new. We distribute directly for the publisher. The Arthur-Selberg trace formula is an equality between two kinds of traces: the geometric terms given by the conjugacy ... classes of a group and the spectral terms given by the induced representations. In general, these terms require a truncation in order to converge, which leads to an equality of truncated kernels. The formulas are difficult in general and even the case of $GL$(2) is nontrivial. The book gives proof of Arthur's trace formula of the 1970s and 1980s, with special attention given to $GL$(2). The problem is that when the truncated terms converge, they are also shown to be polynomial in the truncation variable and expressed as "weighted" orbital and "weighted" characters. In some important cases the trace formula takes on a simple form over $G$. The author gives some examples of this, and also some examples of Jacquet's relative trace formula.This work offers for the first time a simultaneous treatment of a general group with the case of $GL$(2). It also treats the trace formula with the example of Jacquet's relative formula. Read more Show Less

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Overview

The Arthur-Selberg trace formula is an equality between two kinds of traces: the geometric terms given by the conjugacy classes of a group and the spectral terms given by the induced representations. In general, these terms require a truncation in order to converge, which leads to an equality of truncated kernels. The formulas are difficult in general and even the case of $GL$(2) is nontrivial. The book gives proof of Arthur's trace formula of the 1970s and 1980s, with special attention given to $GL$(2). The problem is that when the truncated terms converge, they are also shown to be polynomial in the truncation variable and expressed as ''weighted'' orbital and ''weighted'' characters. In some important cases the trace formula takes on a simple form over $G$. The author gives some examples of this, and also some examples of Jacquet's relative trace formula. This work offers for the first time a simultaneous treatment of a general group with the case of $GL$(2). It also treats the trace formula with the example of Jacquet's relative formula. Features: Discusses why the terms of the geometric and spectral type must be truncated, and why the resulting truncations are polynomials in the truncation of value $T$. Brings into play the significant tool of ($G, M$) families and how the theory of Paley-Weiner is applied. Explains why the truncation formula reduces to a simple formula involving only the elliptic terms on the geometric sides with the representations appearing cuspidally on the spectral side (applies to Tamagawa numbers). Outlines Jacquet's trace formula and shows how it works for $GL$(2).

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Editorial Reviews

Booknews
A book/disk supplementary text for courses in food science and food technology, covering the basics of spreadsheet use by way of examples and problems from areas such as food microbiology, food chemistry, sensory evaluation, statistical quality control, and food engineering. Each problem is presented with equations and instructions. The companion disk contains worked example spreadsheets, and a database of the composition of some 2,400 foods. Assumes little background in mathematics and computers. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780821805718
  • Publisher: American Mathematical Society
  • Publication date: 8/20/1996
  • Series: University Lecture Series, #9
  • Edition number: 1
  • Pages: 99
  • Product dimensions: 6.80 (w) x 9.90 (h) x 0.30 (d)

Table of Contents

Preface
Lecture I Introduction to the Trace Formula 1
Lecture II Arthur's Modified Kernels I: The Geometric Terms 7
Lecture III Arthur's Modified Kernels II: The Spectral Terms 17
Lecture IV More Explicit Forms of the Trace Formula 31
Lecture V Simple Forms of the Trace Formula 45
Lecture VI Applications of the Trace Formula 53
Lecture VII (G, M)-Families and the Spectral J[subscript x](f) 63
Lecture VIII Jacquet's Relative Trace Formula 75
Lecture IX Applications of Paley-Wiener, and Concluding Remarks 87
References 97
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