Epilogue: Formal-Analytic Arithmetic Surfaces and Algebraization

2020 ◽  
pp. 237-303
Author(s):  
Jean-Benoît Bost
Keyword(s):  
Arithmetic Surfaces

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 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.

Analysis on arithmetic schemes. II

2010 ◽  
Vol 5 (3) ◽  
pp. 437-557 ◽  
Author(s):  
Ivan Fesenko
Keyword(s):  
Zeta Function ◽  
Logarithmic Derivative ◽  
Boundary Function ◽  
Boundary Integral ◽  
Integration Theory ◽  
Meromorphic Continuation ◽  
Two Dimensional ◽  
Fundamental Properties ◽  
Arithmetic Surfaces ◽  
Dimensional Version

AbstractWe construct adelic objects for rank two integral structures on arithmetic surfaces and develop measure and integration theory, as well as elements of harmonic analysis. Using the topological Milnor K2-delic and K1×K1-delic objects associated to an arithmetic surface, an adelic zeta integral is defined. Its unramified version is closely related to the square of the zeta function of the surface. For a proper regular model of an elliptic curve over a global field, a two-dimensional version of the theory of Tate and Iwasawa is derived. Using adelic analytic duality and a two-dimensional theta formula, the study of the zeta integral is reduced to the study of a boundary integral term. The work includes first applications to three fundamental properties of the zeta function: its meromorphic continuation and functional equation and a hypothesis on its mean periodicity; the location of its poles and a hypothesis on the permanence of the sign of the fourth logarithmic derivative of a boundary function; and its pole at the central point where the boundary integral explicitly relates the analytic and arithmetic ranks.

Algebraic K-Theory and Classfield Theory for Arithmetic Surfaces

1981 ◽  
Vol 114 (2) ◽  
pp. 229 ◽  
Author(s):  
Spencer Bloch
Keyword(s):  
Arithmetic Surfaces ◽  
K Theory
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