Rankin-Cohen Pairings and Hilbert Hecke Eigenform Product Identities

Abstract

We establish a Petersson-Rankin-Selberg identity for Hilbert Rankin-Cohen brackets over totally real fields of degree greater than one and arbitrary narrow class number. This extends results of Zhang-Zhang and Zhang-Zhou. For even $k,\ell\ge 2$, $r\in\mathbb{Z}_{\ge 0}$, $K=k+\ell+2r$, and normalized full-level cuspidal Hecke eigentuples $f\in S_K(ω_f)$ and $h\in S_\ell(ω_h)$, we show $\langle f,[E_k,h]_r\rangle=C_{F,k,\ell,r}L^S(k/2,Π(f)\timesΠ(h)^\vee)/L_F(k,ω_fω_h^{-1})\ne 0$, where $C_{F,k,\ell,r}>0$. Specializing to $k=2$, we remove the remaining GRH assumption in the product classification of Hao-Qin-Zhou. Hence, over real quadratic fields of narrow class number one, the only full-level product identities among Hecke eigenforms of even parallel weights at least two, up to interchanging the factors, are $E_4^{\mathrm{eig}}=60(E_2^{\mathrm{eig}})^2$ and $h_8=120E_2^{\mathrm{eig}}h_6$ over $\mathbb{Q}(\sqrt{5})$.