Differential operators on Hilbert modular forms

Abstract

We investigate differential operators and their compatibility with subgroups of SL 2 (R) n . In particular, we construct Rankin-Cohen brackets for Hilbert modular forms, and more generally, multilinear differential operators on the space of Hilbert modular forms. As an application, we explicitly determine the RankinCohen bracket of a Hilbert-Eisenstein series and an arbitrary Hilbert modular form. We use this result to compute the Petersson inner product of such a bracket and a Hilbert modular cusp form.

Periods of Hilbert Modular forms, Kronecker series and Cohomology

“…) is the "Kuznetsov lifting" of the Hilbert Eisenstein series G F,k (τ ). Its modular transformation property is known (see Theorem 2 in [6]) : for any (…”

Kernels for products of Hilbert L-functions

“…The following gives a relation between Rankin-Cohen brackets and a double Eisenstein series. Rankin-Cohen brackets on spaces of Hilbert modular forms have been studied in [1]. Let us recall the definition of Rankin-Cohen brackets:…”

“…For part(1), apply the proof of Lemma 4.1 in[5] for each component and we haveN(c γδ −1 ) ≤ N(Im(γz)) −1/2 N(Im(δz)) −1/2 , for any γ, δ ∈ Γ with c γδ −1 ≫ 0. Let r = max{Re(w), 1}.…”

Restrictions of Hilbert Modular Forms

“…Let K be a real quadratic field. We denote by ([·, ·] (p1,p2) ) the R-C operators defined on tensor products of spaces of Hilbert modular forms on Γ K (see [2,8] for the explicit definitions). We have a map:…”