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4-manifold adic algebraic varieties Amer analysis analytic applications asymptotic authors automorphic automorphic forms Banach space boundary bounded bundle Chapter classical coefficients cohomology compact complex computation conjecture construction corresponding curves D-branes defined denote differential equations dimension domain Drinfel'd modules dynamics elliptic English summary example existence extension field theory finite formula function function field functor geometry given global Hilbert space holomorphic homology homology manifolds hyperbolic inequality integral invariant isomorphic L-functions lattice Lie algebra linear M-theory main result manifold Math mathematical matrix method metric modular modular forms module moduli space nonlinear normal obtained operator paper under review perturbation Phys polynomial problem proof properties proved quantum quasi-isometry representation Riemann sequence singularities smooth solutions spectral string theory structure subgroup subset subspace supergravity surface symmetric symplectic theorem topological varieties vector field zero