  The notions of Bloch wave, crystal momentum, and energy bands are commonly
regarded as unique features of crystalline materials with commutative
translation symmetries. Motivated by the recent realization of hyperbolic
lattices in circuit quantum electrodynamics, we exploit ideas from algebraic
geometry to construct the first hyperbolic generalization of Bloch theory,
despite the absence of commutative translation symmetries. For a quantum
particle propagating in a hyperbolic lattice potential, we construct a
continuous family of eigenstates that acquire Bloch-like phase factors under a
discrete but noncommutative group of hyperbolic translations, the Fuchsian
group of the lattice. A hyperbolic analog of crystal momentum arises as the set
of Aharonov-Bohm phases threading the cycles of a higher-genus Riemann surface
associated with this group. This crystal momentum lives in a higher-dimensional
Brillouin zone torus, the Jacobian of the Riemann surface, over which a
discrete set of continuous energy bands can be computed.
