Numerical Computations of Green's Function and Its Fourier Coefficients on PSL(2, ℤ)

Avelin, Helen

doi:10.1080/10586458.2010.10390627

Cited by 4 publications

(2 citation statements)

References 10 publications

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“…For example Hejhal's algorithm can be used to compute embedded eigenvalues with extreme accuracy (see for example [5], see also [4]), and is used to compute large numbers of high lying eigenvalues (for example [22] for arithmetic examples). Variations have also been used to track resonances (for example [9,1,3]). Our approach is di ferent in that it treats the compact part as a black-box and also allows for perturbations away from constant curvature.…”

Section: Plan Of the Paper And Discussion Of The Resultsmentioning

confidence: 99%

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Computations of Eigenvalues and Resonances on Perturbed Hyperbolic Surfaces with Cusps

Levitan

Strohmaier

2019

International Mathematics Research Notices

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In this paper we describe a simple method that allows for a fast direct computation of the scattering matrix for a surface with hyperbolic cusps from the Neumann-to-Dirichlet map on the compact manifold with boundary obtained by removing the cusps. We illustrate that even if the Neumann-to-Dirichlet map is obtained by a Finite Element Method (FEM) one can achieve good accuracy for the scattering matrix. We give various interesting examples of how this can be used to investigate the behaviour of resonances under conformal perturbations or when moving in Teichmüller space. For example, based on numerical experiments we rediscover the four arithmetic surfaces of genus one with one cusp. This demonstrates that it is possible to identify arithmetic objects using FEM.

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Section: Plan Of the Paper And Discussion Of The Resultsmentioning

confidence: 99%

“…The above discussion also shows that the discrete spectrum consists of several parts, each belonging to mixed Dirichlet-Neumann problems on certain domains. Figure 13 shows the computed resonances for = arccosh (3).…”

Section: = Arccosh(3) ≈ 1762747mentioning

confidence: 99%

Computations of Eigenvalues and Resonances on Perturbed Hyperbolic Surfaces with Cusps

Levitan

Strohmaier

2019

International Mathematics Research Notices

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Computations of Green's Function and Its Fourier Coefficients on Fuchsian Groups

Avelin¹

2010

Experimental Mathematics

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Computation of Harmonic Weak Maass Forms

Bruinier

Strömberg

2012

Experimental Mathematics

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Harmonic weak Maass forms of half-integral weight are the subject of many recent works. They are closely related to Ramanujan's mock theta functions, their theta lifts give rise to Arakelov Green functions, and their coefficients are often related to central values and derivatives of Hecke L-functions. We present an algorithm to compute harmonic weak Maass forms numerically, based on the automorphy method due to Hejhal and Stark.As explicit examples we consider harmonic weak Maass forms of weight 1/2 associated to the elliptic curves 11a1, 37a1, 37b1. We made extensive numerical computations and the data we obtained is presented in the final section of the paper. We expect that experiments based on our data will lead to a better understanding of the arithmetic properties of the Fourier coefficients.

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Numerical Computations of Green's Function and Its Fourier Coefficients on PSL(2, ℤ)

Cited by 4 publications

References 10 publications

Computations of Eigenvalues and Resonances on Perturbed Hyperbolic Surfaces with Cusps

Computations of Eigenvalues and Resonances on Perturbed Hyperbolic Surfaces with Cusps

Computations of Green's Function and Its Fourier Coefficients on Fuchsian Groups

Computation of Harmonic Weak Maass Forms

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