The Rankin-Selberg method on congruence subgroups

An explicit arithmetic formula for the Fourier coefficients of Siegel–Eisenstein series of degree two and square-free odd levels

“…The main tools are the functional equation (Proposition 8 in Sect. 4), the KatokSarnak correspondence for Maass wave forms [9,22] and the Rankin-Selberg method for automorphic forms which are not of rapid decay [10,36,44]. In order to state the correspondence, we introduce Maass wave forms of weight 1/2.…”

“…Let us apply the Rankin-Selberg method for automorphic forms which are not of rapid decay [10,36,44] in order to get a suitable Dirichlet series expression for the Koecher-Maass series D * (E (2) k,χ , U, s). Using the same notation in [36] p. 5, some calculations given in [35] imply …”

“…Note that, since involved modular forms are not always cuspidal depending on Maass forms U(τ ), we cannot use the usual Rankin-Selberg method. Instead, one can use Zagier's RankinSelberg method [10,36,44] in this point. In Sect.…”

On Euler products associated with noncuspidal metaplectic forms

“…Zagier's method for functions not of rapid decay has been widely used to handle convolutions which involve metaplectic Eisenstein series and theta series, as they are not cuspidal and their Fourier coefficient contains number theoretic information [11], [15]. Clear formulations and rigorous proofs of the method in contexts other than Zagier's may be found in [13], [3], [4].…”

Renormalization of integrals of products of Eisenstein series and analytic continuation of representations

“…Our initial hope was that at this stage we would be able to use the results of [3] regarding G-invariant Sobolev norms in order to improve the error term in Corollary 2 to O M subgroups and the adelic setting have been carried out by a number of authors, cf. [2,8,9,21]. Here, however, we deal with arbitrary (in particular, non-arithmetic) Γ < PSL(2, R).…”