Algebra of Hyperbolic Band Theory under Magnetic Field

We explore algebras associated with the hyperbolic band theory under a magnetic field for first time. define Fuchsian group higher genus Riemann surface. By imposing boundary conditions Bloch states, we construct states and investigate their energy spectrum. give connection between such automorphic forms. Our is general extension of conventional algebra defined on Euclidean lattice/space into that lattice/Riemann