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Lectures on the Arthur-Selberg Trace Formula

Stephen Gelbart Weizemann Institute of Science, Rehovot, Israel
Available Formats:
Electronic ISBN: 978-1-4704-2158-8
Product Code: ULECT/9.E
List Price: $25.00 MAA Member Price:$22.50
AMS Member Price: $20.00 Click above image for expanded view Lectures on the Arthur-Selberg Trace Formula Stephen Gelbart Weizemann Institute of Science, Rehovot, Israel Available Formats:  Electronic ISBN: 978-1-4704-2158-8 Product Code: ULECT/9.E  List Price:$25.00 MAA Member Price: $22.50 AMS Member Price:$20.00
• Book Details

University Lecture Series
Volume: 91996; 99 pp
MSC: Primary 22; Secondary 11;

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).

Graduate students and professional mathematicians interested in topological groups and Lie groups.

• Chapters
• Lecture I. Introduction to the trace formula
• Lecture II. Arthur’s modified kernels I: The geometric terms
• Lecture III. Arthur’s modified kernels II: The spectral terms
• Lecture IV. More explicit forms of the trace formula
• Lecture V. Simple forms of the trace formula
• Lecture VI. Applications of the trace formula
• Lecture VII. $(G,M)$-families and the spectral $J_{\chi }(f)$
• Lecture VII. Jacquet’s relative trace formula
• Lecture IX. Applications of Paley–Wiener, and concluding remarks
• Reviews

• For graduate students or for seasoned researchers wishing to expand their repertoire, this book will be invaluable. Many basic facts … are collected in one convenient reference.

Mathematical Reviews
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Volume: 91996; 99 pp
MSC: Primary 22; Secondary 11;

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).

Graduate students and professional mathematicians interested in topological groups and Lie groups.

• Chapters
• Lecture I. Introduction to the trace formula
• Lecture II. Arthur’s modified kernels I: The geometric terms
• Lecture III. Arthur’s modified kernels II: The spectral terms
• Lecture IV. More explicit forms of the trace formula
• Lecture V. Simple forms of the trace formula
• Lecture VI. Applications of the trace formula
• Lecture VII. $(G,M)$-families and the spectral $J_{\chi }(f)$
• Lecture VII. Jacquet’s relative trace formula
• Lecture IX. Applications of Paley–Wiener, and concluding remarks
• For graduate students or for seasoned researchers wishing to expand their repertoire, this book will be invaluable. Many basic facts … are collected in one convenient reference.

Mathematical Reviews
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