Arakelov geometry on arithmetic surfaces

2014 ◽  
pp. 87-140
Keyword(s):  
Arithmetic Surfaces ◽  
Arakelov Geometry

scholarly journals A tropical approach to nonarchimedean Arakelov geometry

2017 ◽  
Vol 11 (1) ◽  
pp. 77-180 ◽  
Author(s):  
Walter Gubler ◽  
Klaus Künnemann
Keyword(s):  
Arakelov Geometry

scholarly journals Arithmetic surfaces and adelic quotient groups

2018 ◽  
Vol 82 (4) ◽  
pp. 817-836 ◽  
Author(s):  
D. V. Osipov
Keyword(s):  
Arithmetic Surfaces ◽  
Quotient Groups

scholarly journals The genus two free boson in Arakelov geometry

2020 ◽  
Vol 960 ◽  
pp. 115184
Author(s):  
Thomas Vandermeulen
Keyword(s):  
Free Boson ◽  
Genus Two ◽  
Arakelov Geometry

scholarly journals Computations of Eigenvalues and Resonances on Perturbed Hyperbolic Surfaces with Cusps

2019 ◽  
Author(s):  
Michael Levitin ◽  
Alexander Strohmaier
Keyword(s):  
Finite Element Method ◽  
Numerical Experiments ◽  
Compact Manifold ◽  
Good Accuracy ◽  
Scattering Matrix ◽  
Direct Computation ◽  
Simple Method ◽  
Manifold With Boundary ◽  
Arithmetic Surfaces ◽  
Genus One

Abstract 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. All the videos accompanying this paper are available with its online version, or externally either at michaellevitin.net/hyperbolic.html or as a dedicated YouTubeplaylist.

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