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Author: Salahoddin Shokranian Publisher: Springer ISBN: 3540466592 Category : Mathematics Languages : en Pages : 104
Book Description
This book based on lectures given by James Arthur discusses the trace formula of Selberg and Arthur. The emphasis is laid on Arthur's trace formula for GL(r), with several examples in order to illustrate the basic concepts. The book will be useful and stimulating reading for graduate students in automorphic forms, analytic number theory, and non-commutative harmonic analysis, as well as researchers in these fields. Contents: I. Number Theory and Automorphic Representations.1.1. Some problems in classical number theory, 1.2. Modular forms and automorphic representations; II. Selberg's Trace Formula 2.1. Historical Remarks, 2.2. Orbital integrals and Selberg's trace formula, 2.3.Three examples, 2.4. A necessary condition, 2.5. Generalizations and applications; III. Kernel Functions and the Convergence Theorem, 3.1. Preliminaries on GL(r), 3.2. Combinatorics and reduction theory, 3.3. The convergence theorem; IV. The Ad lic Theory, 4.1. Basic facts; V. The Geometric Theory, 5.1. The JTO(f) and JT(f) distributions, 5.2. A geometric I-function, 5.3. The weight functions; VI. The Geometric Expansionof the Trace Formula, 6.1. Weighted orbital integrals, 6.2. The unipotent distribution; VII. The Spectral Theory, 7.1. A review of the Eisenstein series, 7.2. Cusp forms, truncation, the trace formula; VIII.The Invariant Trace Formula and its Applications, 8.1. The invariant trace formula for GL(r), 8.2. Applications and remarks
Author: Salahoddin Shokranian Publisher: Springer ISBN: 3540466592 Category : Mathematics Languages : en Pages : 104
Book Description
This book based on lectures given by James Arthur discusses the trace formula of Selberg and Arthur. The emphasis is laid on Arthur's trace formula for GL(r), with several examples in order to illustrate the basic concepts. The book will be useful and stimulating reading for graduate students in automorphic forms, analytic number theory, and non-commutative harmonic analysis, as well as researchers in these fields. Contents: I. Number Theory and Automorphic Representations.1.1. Some problems in classical number theory, 1.2. Modular forms and automorphic representations; II. Selberg's Trace Formula 2.1. Historical Remarks, 2.2. Orbital integrals and Selberg's trace formula, 2.3.Three examples, 2.4. A necessary condition, 2.5. Generalizations and applications; III. Kernel Functions and the Convergence Theorem, 3.1. Preliminaries on GL(r), 3.2. Combinatorics and reduction theory, 3.3. The convergence theorem; IV. The Ad lic Theory, 4.1. Basic facts; V. The Geometric Theory, 5.1. The JTO(f) and JT(f) distributions, 5.2. A geometric I-function, 5.3. The weight functions; VI. The Geometric Expansionof the Trace Formula, 6.1. Weighted orbital integrals, 6.2. The unipotent distribution; VII. The Spectral Theory, 7.1. A review of the Eisenstein series, 7.2. Cusp forms, truncation, the trace formula; VIII.The Invariant Trace Formula and its Applications, 8.1. The invariant trace formula for GL(r), 8.2. Applications and remarks
Author: Stephen S. Gelbart Publisher: American Mathematical Soc. ISBN: 0821805711 Category : Selberg trace formula Languages : en Pages : 112
Book Description
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).
Author: M. Scott Osborne Publisher: American Mathematical Soc. ISBN: 0821822837 Category : Automorphic forms Languages : en Pages : 217
Book Description
In this memoir, we lay the foundations for the study of inner product formulae, one of the key technical preliminaries in the derivation of the Selberg trace formula.
Author: Isaac Y. Efrat Publisher: American Mathematical Soc. ISBN: 0821824244 Category : Mathematics Languages : en Pages : 121
Book Description
We evaluate the Selberg trace formula for all discrete, irreducible, cofinite subgroups of PSL2 ([double-struck capital]R)[italic superscript]n. In particular, this involves studying the spectral theory of the fundamental domain, and the analysis of the appropriate Eisenstein series. A special role is played by the Hilbert modular groups, both because of their relation to the general case, stemming from a rigidity theorem, and their inherent algebraic number theoretic interest.
Author: Salahoddin Shokranian Publisher: Springer ISBN: 9783540550211 Category : Mathematics Languages : en Pages : 116
Book Description
This book based on lectures given by James Arthur discusses the trace formula of Selberg and Arthur. The emphasis is laid on Arthur's trace formula for GL(r), with several examples in order to illustrate the basic concepts. The book will be useful and stimulating reading for graduate students in automorphic forms, analytic number theory, and non-commutative harmonic analysis, as well as researchers in these fields. Contents: I. Number Theory and Automorphic Representations.1.1. Some problems in classical number theory, 1.2. Modular forms and automorphic representations; II. Selberg's Trace Formula 2.1. Historical Remarks, 2.2. Orbital integrals and Selberg's trace formula, 2.3.Three examples, 2.4. A necessary condition, 2.5. Generalizations and applications; III. Kernel Functions and the Convergence Theorem, 3.1. Preliminaries on GL(r), 3.2. Combinatorics and reduction theory, 3.3. The convergence theorem; IV. The Ad lic Theory, 4.1. Basic facts; V. The Geometric Theory, 5.1. The JTO(f) and JT(f) distributions, 5.2. A geometric I-function, 5.3. The weight functions; VI. The Geometric Expansionof the Trace Formula, 6.1. Weighted orbital integrals, 6.2. The unipotent distribution; VII. The Spectral Theory, 7.1. A review of the Eisenstein series, 7.2. Cusp forms, truncation, the trace formula; VIII.The Invariant Trace Formula and its Applications, 8.1. The invariant trace formula for GL(r), 8.2. Applications and remarks
Author: Salahoddin Shokranian
Author: Salahoddin Shokranian
Author: Salahoddin Shokranian
Author: Salahoddin Shokranian
Author: Stephen S. Gelbart
Author: Dennis A. Hejhal
Author: Laurent Clozel
Author: M. Scott Osborne
Author: Dennis A. Hejhal
Author: Isaac Y. Efrat
Author: Salahoddin Shokranian