Using the theta correspondence, we study a lift from (not necessarily rapidly decreasing) closed differential (p−n)-forms on a non-compact arithmetic quotient of hyperbolic p-space to Siegel modular forms of degree n. This generalizes earlier work of Kudla and the second named author (in the case of hyperbolic space). We give a cohomological interpretation of the lift and analyze its Fourier expansion in terms of periods over certain cycles. For Riemann surfaces, i.e., the case p= 2, we obtain a complete description using the theory of Eisenstein cohomology.
Funke, J., & Millson, J. (2002). Cycles in hyperbolic manifolds of non-compact type and Fourier coefficients of Siegel modular forms. manuscripta mathematica, 107(4), 409-449. https://doi.org/10.1007/s002290100241