Relative Trace Formulas

Relative Trace Formulas PDF

Author: Werner Müller

Publisher: Springer Nature

Published: 2021-05-18

Total Pages: 438

ISBN-13: 3030685063

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A series of three symposia took place on the topic of trace formulas, each with an accompanying proceedings volume. The present volume is the third and final in this series and focuses on relative trace formulas in relation to special values of L-functions, integral representations, arithmetic cycles, theta correspondence and branching laws. The first volume focused on Arthur’s trace formula, and the second volume focused on methods from algebraic geometry and representation theory. The three proceedings volumes have provided a snapshot of some of the current research, in the hope of stimulating further research on these topics. The collegial format of the symposia allowed a homogeneous set of experts to isolate key difficulties going forward and to collectively assess the feasibility of diverse approaches.

Families of Automorphic Forms and the Trace Formula

Families of Automorphic Forms and the Trace Formula PDF

Author: Werner Müller

Publisher: Springer

Published: 2016-09-20

Total Pages: 581

ISBN-13: 3319414240

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Featuring the work of twenty-three internationally-recognized experts, this volume explores the trace formula, spectra of locally symmetric spaces, p-adic families, and other recent techniques from harmonic analysis and representation theory. Each peer-reviewed submission in this volume, based on the Simons Foundation symposium on families of automorphic forms and the trace formula held in Puerto Rico in January-February 2014, is the product of intensive research collaboration by the participants over the course of the seven-day workshop. The goal of each session in the symposium was to bring together researchers with diverse specialties in order to identify key difficulties as well as fruitful approaches being explored in the field. The respective themes were counting cohomological forms, p-adic trace formulas, Hecke fields, slopes of modular forms, and orbital integrals.

Lectures on the Arthur-Selberg Trace Formula

Lectures on the Arthur-Selberg Trace Formula PDF

Author: Stephen S. Gelbart

Publisher: American Mathematical Soc.

Published: 1996

Total Pages: 112

ISBN-13: 0821805711

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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).

A Local Relative Trace Formula for the Ginzburg-Rallis Model: The Geometric Side

A Local Relative Trace Formula for the Ginzburg-Rallis Model: The Geometric Side PDF

Author: Chen Wan

Publisher: American Mathematical Soc.

Published: 2019-12-02

Total Pages: 90

ISBN-13: 1470436868

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Following the method developed by Waldspurger and Beuzart-Plessis in their proofs of the local Gan-Gross-Prasad conjecture, the author is able to prove the geometric side of a local relative trace formula for the Ginzburg-Rallis model. Then by applying such formula, the author proves a multiplicity formula of the Ginzburg-Rallis model for the supercuspidal representations. Using that multiplicity formula, the author proves the multiplicity one theorem for the Ginzburg-Rallis model over Vogan packets in the supercuspidal case.

Periods and Relative Trace Formulas for GL(2) in the Local Setting

Periods and Relative Trace Formulas for GL(2) in the Local Setting PDF

Author: Brooke Gabrielle Feigon

Publisher:

Published: 2006

Total Pages: 170

ISBN-13: 9780542881657

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By comparing the relative and Kuznetsov trace formulas in the global setting, Jacquet developed a method for characterizing the image of the base change map associating automorphic representations of U(2, AE/AF) to automorphic representations of GL(2, A E). Here we define, prove and compare local versions of the relative and Kuznetsov trace formulas on GL(2) and U(2). When evaluated with matching functions, the local Kuznetsov trace formula and the local relative trace formula are equal and thus there is an equality between their local distributions on the spectral sides. To define the local distributions for the relative trace formula, we define a regularized period integral and prove that it is a GL(2, F) invariant linear functional.

Relative Aspects in Representation Theory, Langlands Functoriality and Automorphic Forms

Relative Aspects in Representation Theory, Langlands Functoriality and Automorphic Forms PDF

Author: Volker Heiermann

Publisher: Springer

Published: 2018-10-01

Total Pages: 364

ISBN-13: 3319952315

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This volume presents a panorama of the diverse activities organized by V. Heiermann and D. Prasad in Marseille at the CIRM for the Chaire Morlet event during the first semester of 2016. It assembles together expository articles on topics which previously could only be found in research papers. Starting with a very detailed article by P. Baumann and S. Riche on the geometric Satake correspondence, the book continues with three introductory articles on distinguished representations due to P. Broussous, F. Murnaghan, and O. Offen; an expository article of I. Badulescu on the Jacquet–Langlands correspondence; a paper of J. Arthur on functoriality and the trace formula in the context of "Beyond Endoscopy", taken from the Simons Proceedings; an article of W-W. Li attempting to generalize Godement–Jacquet theory; and a research paper of C. Moeglin and D. Renard, applying the trace formula to the local Langlands classification for classical groups. The book should be of interest to students as well as professional researchers working in the broad area of number theory and representation theory.

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