Standard compact periods for Eisenstein series

When o has larger class number, linear combinations of CM-point values corresponding to the ideal classes yield the corresponding ratio. This example was understood in the 19th century, and is the simplest in well-known families of special values and periods of Eisenstein series. The next case in order of increasing complexity is that of integrals of the same Es along hyperbolic geodesics in H that have compact images in Γ\H. These were considered by Hecke and Maass. In fact, from a contemporary viewpoint the CM-point value example and the hyperbolic geodesic periods example are identical, as is made clear in the first section, The first three examples are instances of periods of Eisenstein series on orthogonal groups O(n + 1) with rational rank 1, along orthogonal groups O(n) with compact arithmetic quotients. Another family extending the small examples is periods of degenerate Eisenstein series along anisotropic tori. Periods of Eisenstein series are interesting prototypes for periods of cuspforms, often unwinding more completely than the corresponding integrals for cuspforms. In that context, periods attached to degenerate Eisenstein series are inevitably misleading to some degree. Nevertheless, there is a correct indication that sharp estimates on periods are often connected to Lindelöf hypotheses and other very serious issues. See the bibliography for background and some pointers to contemporary literature.