Integral moments of automorphic L-functions

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.