The Selberg Trace Formula for PSL(2,ℝ)

Authors

Dennis A. Hejhal

Type: Book

Publication Date: 1983-01-01

Citations: 858

DOI: https://doi.org/10.1007/bfb0061302

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Lecture notes in mathematics
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Library Union Catalog of Bavaria, Berlin and Brandenburg (B3Kat Repository)
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Summary

This book by Dennis A. Hejhal, “The Selberg Trace Formula for PSL(2,ℝ),” Volume 2, published in 1983, is a detailed treatment of the Selberg trace formula, focusing primarily on Fuchsian groups with at least one cusp. It builds upon the foundations laid in Selberg’s earlier work and the author’s own Volume 1.

Significance:
The Selberg trace formula is a central result in the theory of automorphic forms, connecting spectral data (eigenvalues of the Laplacian) with geometric data (lengths of closed geodesics) on locally symmetric spaces. It is a non-commutative analogue of the Poisson summation formula and the Riemann explicit formula. This formula has far-reaching consequences in number theory, representation theory, and mathematical physics, enabling connections between seemingly disparate areas of mathematics. This volume represents an important effort to provide a clear and accessible exposition of this complex and powerful tool, along with numerous applications.

Key Innovations and Contributions:

Systematic Elaboration: The book offers a systematic and detailed exposition of Selberg’s ideas, particularly his Göttingen notes from 1954, making them more accessible to a wider audience.
Treatment of Real Weight: It carefully develops the trace formula for automorphic forms of arbitrary real weight, which is a crucial extension.
Alternative Derivation: An alternate (and earlier) approach of Selberg for continuing the Eisenstein series and deriving the trace formula is published for the first time.
Applications: It explores applications of the trace formula to various problems, including the dimension of spaces of holomorphic automorphic forms, concrete examples for classical Fuchsian groups and numerical computations of eigenvalues.
Appendices: Includes appendices covering supplementary materials such as Kloosterman sums and a detailed exposition of Selberg’s third method for analytically continuing Eisenstein series.

Main Prior Ingredients Needed:

Basic Knowledge of Fuchsian Groups: Understanding of the theory of Fuchsian groups, including their fundamental regions, hyperbolic geometry, and group actions on the upper half-plane is essential.
Spectral Theory of the Laplacian: Familiarity with the spectral theory of the Laplacian operator on hyperbolic spaces, including eigenvalues, eigenfunctions, and the resolvent kernel.
Theory of Automorphic Forms: A solid grounding in the theory of automorphic forms, including Eisenstein series, cusp forms, and their properties, is necessary.
Selberg’s Earlier Work: Prior knowledge of Selberg’s earlier work on trace formulas and zeta functions is highly beneficial, particularly his Göttingen notes and Volume 1.
Functional Analysis and Complex Analysis: Strong background in functional analysis and complex analysis, especially the theory of meromorphic continuation, is needed to understand the analytical aspects of the trace formula.

In essence, Hejhal’s Volume 2 serves as a comprehensive and detailed guide to the Selberg trace formula for PSL(2,ℝ), bridging the gap between Selberg’s original work and the broader mathematical community.

The Selberg Trace Formula for PSL (2,R): Volume 2 (Lecture Notes in Mathematics (1001))

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Dennis A. Hejhal

Book Review: The Selberg trace formula for $PSL\,\left( {2,\mathbf{R}} \right)$, Volume I

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THE SELBERG TRACE FORMULA FOR PSL(2, <i>R</i> ) Volume 2: (Lecture Notes in Mathematics, 1001)

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S. J. Patterson

By Dennis A. Hejhal: pp. 806. DM.89–; US$35.40. (Springer-Verlag, Berlin, 1983.) By Dennis A. Hejhal: pp. 806. DM.89–; US$35.40. (Springer-Verlag, Berlin, 1983.)

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Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the homogeneous space associated to the group <inline-formula … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the homogeneous space associated to the group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="PGL Subscript 3 Baseline left-parenthesis bold upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mtext>PGL</mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>3</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\text {PGL}_{3}(\mathbf {R})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X equals normal upper Gamma negative script upper H"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-variant" mathvariant="normal">∖</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">X = \Gamma {\backslash \mathcal {H}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma equals SL Subscript 3 Baseline left-parenthesis bold upper Z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mtext>SL</mml:mtext> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>3</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\Gamma = \text {SL}_{3}(\mathbf {Z})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and consider the first nontrivial eigenvalue <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda 1"> <mml:semantics> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\lambda _{1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the Laplacian on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L^{2}(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Using geometric considerations, we prove the inequality <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda 1 greater-than 3 pi squared slash 10"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>></mml:mo> <mml:mn>3</mml:mn> <mml:msup> <mml:mi>π</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>10</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\lambda _{1} > 3\pi ^{2}/10</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Since the continuous spectrum is represented by the band <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 1 comma normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[1,\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, our bound on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda 1"> <mml:semantics> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\lambda _{1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be viewed as an analogue of Selberg’s eigenvalue conjecture for quotients of the hyperbolic half space.

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2014-08-04

Anke Pohl

By a transfer operator approach to Maass cusp forms and the Selberg zeta function for cofinite Hecke triangle groups, Möller and the present author found a factorization of the Selberg … By a transfer operator approach to Maass cusp forms and the Selberg zeta function for cofinite Hecke triangle groups, Möller and the present author found a factorization of the Selberg zeta function into a product of Fredholm determinants of transfer-operator-like families: $$\begin{eqnarray}Z(s)=\det (1-{\mathcal{L}}_{s}^{+})\det (1-{\mathcal{L}}_{s}^{-}).\end{eqnarray}$$ In this article we show that the operator families ${\mathcal{L}}_{s}^{\pm }$ arise as families of transfer operators for the triangle groups underlying the Hecke triangle groups, and that for $s\in \mathbb{C}$ , $\text{Re}s={\textstyle \frac{1}{2}}$ , the operator ${\mathcal{L}}_{s}^{+}$ (respectively ${\mathcal{L}}_{s}^{-}$ ) has a 1-eigenfunction if and only if there exists an even (respectively odd) Maass cusp form with eigenvalue $s(1-s)$ . For non-arithmetic Hecke triangle groups, this result provides a new formulation of the Phillips–Sarnak conjecture on non-existence of even Maass cusp forms.

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A linear lower bound on the gonality of modular curves

1996-01-01

Dan Abramovich

The result in the title is proven, using the Selberg estimate on the leading eigenvalue of the non-Euclidean Laplacian, and the method of conformal volumes of Li and Yau. The result in the title is proven, using the Selberg estimate on the leading eigenvalue of the non-Euclidean Laplacian, and the method of conformal volumes of Li and Yau.

The Singular Theta Correspondence, Lorentzian Lattices and Borcherds-Kac-Moody Algebras

2003-01-01

Alex V. Barnard

This dissertation answers some of the questions raised in Borcherds' papers on Moonshine and Lorentzian reflection groups. We prove (assuming an open conjecture of Burger, Li and Sarnak) that a … This dissertation answers some of the questions raised in Borcherds' papers on Moonshine and Lorentzian reflection groups. We prove (assuming an open conjecture of Burger, Li and Sarnak) that a Lorentzian reflection group with Weyl vector is associated to a vector-valued modular form. This result allows us to establish a folklore conjecture that the maximal dimension of a Lorentzian reflection group with Weyl vector is 26. In the case of elementary lattices, we show that these vector-valued forms can be obtained by inducing scalar-valued forms. This allows us to explain the critical signatures which occur in Borcherds' work. Many of the structures which occur at these critical signatures are especially beautiful and have appeared independently throughout the literature. We investigate Borcherds-Kac-Moody (BKM) algebras with denominator formulas that are singular weight automorphic forms. The results of these investigations suggest that all such BKM algebras are related to orbifold constructions of vertex algebras and elements of the Monster sporadic group. Finally, we show how work in this dissertation combined with results of Bruinier gives a new insight into the arithmetic mirror symmetry conjecture of Gritsenko and Nikulin.

Density results for automorphic forms on Hilbert modular groups

2003-08-01

Roelof W. Bruggeman , Roberto J. Miatello , I. Pacharoni

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The prime geodesic theorem in square mean

2018-05-18

Antal Balog , András Bíró , Gergely Harcos +1 more

We strengthen the recent result of Cherubini and Guerreiro on the square mean of the error term in the prime geodesic theorem for $\mathrm{PSL}_2(\mathbb{Z})$. We also develop a short interval … We strengthen the recent result of Cherubini and Guerreiro on the square mean of the error term in the prime geodesic theorem for $\mathrm{PSL}_2(\mathbb{Z})$. We also develop a short interval version of this result.

Short proof of Rademacher’s formula for partitions

2019-02-22

Wladimir de Azevedo Pribitkin , Brandon Williams

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LOCAL AVERAGE OF THE HYPERBOLIC CIRCLE PROBLEM FOR FUCHSIAN GROUPS

2018-01-01

András Bíró

Let be a finite-volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the -orbit of in a hyperbolic circle around of radius , … Let be a finite-volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the -orbit of in a hyperbolic circle around of radius , where and are given points of the upper half plane and is a large number. An estimate with error term is known, and this has not been improved for any group. Recently, Risager and Petridis proved that in the special case taking and averaging over in a certain way the error term can be improved to . Here we show such an improvement for a general ; our error term is (which is better than but weaker than the estimate of Risager and Petridis in the case ). Our main tool is our generalization of the Selberg trace formula proved earlier.

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Applications of Guinand’s Formula

1987-01-01

P. X. Gallagher

The explicit formula of Weil [21] connects quite general sums over primes with corresponding sums over the critical zeros of the Riemann zeta function (or more general L-functions). In the … The explicit formula of Weil [21] connects quite general sums over primes with corresponding sums over the critical zeros of the Riemann zeta function (or more general L-functions). In the earlier version of Guinand [8], there is on the Riemann hypothesis1) a kind of Fourier duality between the differentials of the remainder terms in the prime number theorem (suitable renormalized) and in the formula counting critical zeros of the Riemann zeta function.

Superconformal Algebras and Mock Theta Functions 2. Rademacher Expansion for K3 Surface

2009-01-01

Tohru Eguchi , Kazuhiro Hikami

The elliptic genera of the K3 surfaces, both compact and non-compact cases, are studied by using the theory of mock theta functions. We decompose the elliptic genus in terms of … The elliptic genera of the K3 surfaces, both compact and non-compact cases, are studied by using the theory of mock theta functions. We decompose the elliptic genus in terms of the N=4 superconformal characters at level-1, and present an exact formula for the coefficients of the massive (non-BPS) representations using the Poincare-Maass series.

None

2001-01-01

R. Aurich , Frank Steiner