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,ℓ≥ 2, r∈ℤ≥ 0, K=k+ℓ+2r, and normalized full-level cuspidal Hecke eigentuples f∈ SK(ωf) and h∈ Sℓ(ωh), we show ⟨ f,[Ek,h]r⟩=CF,k,ℓ,rLS(k/2,Π(f)×Π(h)∨)/LF(k,ωfωh−1)≠ 0, where CF,k,ℓ,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 E4eig=60(E2eig)2 and h8=120E2eigh6 over ℚ(√5).
Related
BibTeX
@misc{li2026rankincohen,
title = {{Rankin-Cohen Pairings and Hilbert Hecke Eigenform Product Identities}},
author = {Jialin Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28315},
eprint = {2609.28315},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28315},
}
RIS
TY - UNPB
TI - Rankin-Cohen Pairings and Hilbert Hecke Eigenform Product Identities
AU - Jialin Li
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28315
UR - https://arxiv.org/abs/2609.28315
AB - 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,ℓ≥ 2, r∈ℤ_≥ 0, K=k+ℓ+2r, and normalized full-level cuspidal Hecke eigentuples f∈ S_K(ω_f) and h∈ S_ℓ(ω_h), we show ⟨ f,[E_k,h]_r⟩=C_F,k,ℓ,rL^S(k/2,Π(f)×Π(h)^∨)/L_F(k,ω_fω_h^−1)≠ 0, where C_F,k,ℓ,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^eig=60(E_2^eig)^2 and h_8=120E_2^eigh_6 over ℚ(√5).
ER -