A remark on the Hilbert modular forms of weight 1

Abstract

An elliptic modular form of positive integral weight k is a linear combination of a cusp form and Eisenstein series. This well-known fact was proved by Hecke and was generalized by Kloostermann to the Hilbert modular case (Hecke [5] and Kloostermann [8]). The case k = 1, which was left unsettled in [-8], was solved by Gundlach under certain conditions; a Hilbert modular form for the full modular group over a real quadratic field is a linear combination of a cusp form and Eisenstein series (Gundlach [3, 4]). The aim of the present paper is to show that the same holds in general for any principal congruence subgroup over a totally real number field. As one expects by [3], the number of linearly independent Eisenstein series of weight 1 is less than (roughly equal to one half of) the number of cusps. It is enough to prove that the number of linearly independent non-cuspidal forms equals that of linearly independent Eisenstein series. To do this, we shall apply the theory of automorphic representations (the equivalence condition, the uniqueness of Whittaker models, etc .... ) as is developed in Jacquet and Langlands [6].