We obtain second integral moments of automorphic L-functions on adele groups GL2 over arbitrary number fields, by a spectral decomposition using the structure and representation theory of adele groups GL1 and GL2. This requires reformulation of the notion of Poincaré series, replacing the collection of classical Poincaré series over GL2(ℚ) or GL2(ℚ(i)) with a single, coherent, global object that makes sense over a number field. This is the first expression of integral moments in adele-group terms, distinguishing global and local issues, and allowing uniform application to number fields. When specialized to the field of rational numbers ℚ, we recover the classical results on moments.
