Regularized periods of automorphic forms on \(\mathrm{GL}(2)\) (Q386803)

Let \(F\) be an algebraic number field and \(\mathbb{A}\) denote its ring of adéles. Let \(\varphi:\mathrm{GL}(2,F)\backslash \mathrm{GL}(2,\mathbb{A})\to \mathbb{C}\) be an automorphic function and \(\eta\) be a unitary idèle class character. Then \textit{M. Tsuzuki} [Mem. Am. Math. Soc. 1110, 129 p. (2015; Zbl 1337.11034)] described a regularized period of \(\varphi\) against \(\eta\), \(P^{\eta}(\varphi)\).NEWLINENEWLINEIf \(F\) is totally real, Tsuzuki showed that the regularized periods of a cusp form \(\varphi\) in the space of an automorphic representation \(\pi\) are explicitly given in terms of the central \(L\)-value \(L(1/2,\pi\otimes\eta)\), while the regularized periods of Eisenstein series are given in terms of Hecke \(L\)-functions attached to the inducing data.NEWLINENEWLINEIn this paper these results on regularized periods are extended in several ways. First, \(F\) is allowed to be an arbitrary number field. Second, in the cuspidal case the level is allowed to be arbitrary, while Tsuzuki restricted to the square-free case. Third, in the Eisenstein case, the level is similarly more general and the inducing data is not assumed to be unramified. These generalizations are achieved by constructing explicit orthogonal bases for the corresponding local spaces and carrying out computations for these vectors of the regularized period.