Cm Number Fields and Modular Forms

x∂F⊂a⊂OF (Na)k−1. Here the superscript ‘+′ stands for totally positive elements. Siegel further derived from this a simpler proof of his famous theorem that ζF (1− k) is rational for all k ≥ 1. Siegel’s construction is based on the simple observation that a Hilbert modular form becomes an elliptic modular form when restricting diagonally to the upper half plane. Indeed, Hecke constructed and proved in 1924 that (1.1) Ek(τ) = ζF (1− k) + 2d−1 ∑ x∈∂−1,+ F σk−1(x∂F )e(trxτ)