Algèbres de restrictions des formes modulaires de Hilbert

Abstract

Let Γ ⊂ PSL ( 2 , R ) be a discrete subgroup of finite covolume. We suppose that the modular curve H / Γ is ‘embedded’ into a Hilbert modular surface H 2 / Γ K , where Γ K is the Hilbert modular group associated to a real quadratic field K. We define a sequence of restrictions ( ρ n ) n ∈ N of Hilbert modular forms for Γ K to modular forms for Γ. We denote by M k 1 , k 2 ( Γ K ) the space of Hilbert modular forms of weight ( k 1 , k 2 ) for Γ K . We prove that ∑ n ∈ N ∑ k 1 , k 2 ∈ N ρ n ( M k 1 , k 2 ( Γ K ) ) is a subspace closed under Rankin–Cohen brackets of the space ⊕ k ∈ N M k ( Γ ) of modular forms for Γ. To cite this article: N. Ouled Azaiez, C. R. Acad. Sci. Paris, Ser. I 344 (2007).