Eisenstein series in the Kohnen plus space for Hilbert modular forms (Q2800153)

Let \(F\) be a totally real number field of class number \(h,\) with ring of integers \(\mathfrak 0\) and different \(\mathfrak d\), and let \(\Gamma\) be the congruence subgroup NEWLINE\[NEWLINE \left\{\left.\begin{pmatrix} a&b\\ c& d \end{pmatrix} \in \mathrm{SL}_2(F) \right|a,d \in \mathfrak o, c \in \mathfrak d^{-1}, b \in 4\mathfrak d \right\}. NEWLINE\]NEWLINE The plus space \(M^+_{\kappa +1/2}(\Gamma)\) was introduced by [\textit{K. Hiraga} and \textit{T. Ikeda} Compos. Math. 149, No. 12, 1963--2010 (2013; Zbl 1368.11044)] following [{W. Kohnen}, Math. Ann. 248, 249--266 (1980; Zbl 0416.10023)]. Here \(\kappa\) is an integer greater than \(1.\)NEWLINENEWLINEHiraga and Ikeda [loc. cit.] also constructed an isomorphism between the subspace of cusp forms and a certain space of cusp forms of weight \(2 \kappa\) for \(\mathrm{PGL}_2,\) and gave a basis consisting of eigenforms of a suitable Hecke algebra.NEWLINENEWLINEThe purpose of the present paper is to give a suitable basis for the space of Eisenstein series. In the case of \(\mathbb Q\) this was accomplished by \textit{H. Cohen} [Sémin. Théor. Nombres 1974--1975, Univ. Bordeaux, Exp. No. 3, 21 p. (1975; Zbl 0316.10015)], with the answer being a quite elegant sum over fundamental discriminants, involving the associated quadratic Dirichlet characters and their \(L\) functions, as well as the Möbius and divisor functions. For a general totally real number field, the codimension of the space of cusp forms is equal to the class number \(h\), and the author obtains \(h\) linearly independent Eisenstein series. These Eisenstein series are given by a natural generalization of Cohen's formula, they are eigenfunctions of the Hecke algebra, and, together with the Hiraga-Ikeda basis for the space of cusp forms they span the plus space \(M^+_{\kappa +1/2}(\Gamma).\)