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