We discuss the algebraic quantization of a real, massive scalar field in the Poincaré patch of the (d+1)-dimensional anti-de Sitter spacetime, with arbitrary boundary conditions. By using the functional formalism, we show that it is always possible to associate to such system an algebra of observables enjoying the standard properties of causality, time-slice axiom and F-locality. In addition, we characterize the wavefront set of the ground state associated to this system. As a consequence, we are able to generalize the definition of Hadamard states and construct a global algebra of Wick polynomials.