RESTRICTIONS OF HILBERT MODULAR FORMS

Abstract

Let Γ ⊂ PSL (2, ℝ) be a discrete and finite covolume subgroup. We suppose that the modular curve [Formula: see text] is "embedded" in a Hilbert modular surface [Formula: see text], where ΓKis the Hilbert modular group associated to a real quadratic field K. We define a sequence of restrictions (ρn)n∈ℕof Hilbert modular forms for ΓKto modular forms for Γ. We denote by Mk1, k2(ΓK) the space of Hilbert modular forms of weight (k1, k2) for ΓK. We prove that ∑n∈ℕ∑k1, k2∈ℕρn(Mk1, k2(ΓK)) is a subalgebra closed under Rankin–Cohen brackets of the algebra ⊕k∈ℕMk(Γ) of modular forms for Γ.