PreprintCC BY 4.0arXiv
Simultaneous nonvanishing of quadratic twists via Rankin-Cohen brackets
Abstract
Let D be an odd fundamental discriminant, with D=1 permitted, and let r≥ 1 be fixed. We prove that, for every sufficiently large integer ℓ satisfying (−1)ℓ D>0, the first r traced diagonal Rankin--Cohen brackets Tr1|D|[Gℓ−2e,D,Gℓ−2e,D]2e, 1≤ e≤ r, are linearly independent in S2ℓ(SL2(ℤ)). Here Gk,D is the Eisenstein series of weight k, level |D|, and nebentypus χD. The Petersson formula of Kayath, Lane, Neifeld, Ni, and Xue then implies that at least r normalized Hecke eigenforms f∈ S2ℓ(SL2(ℤ)) satisfy L(f⊗χD,ℓ)≠ 0. For D=1, this gives, for every fixed r and every sufficiently large K≡ 0pmod 4, at least r level-one eigenforms of weight K with nonzero central value.
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BibTeX
@misc{takloobighash2026simultaneous,
title = {{Simultaneous nonvanishing of quadratic twists via Rankin-Cohen brackets}},
author = {Ramin Takloo-Bighash},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19649},
eprint = {2609.19649},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19649},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Simultaneous nonvanishing of quadratic twists via Rankin-Cohen brackets
AU - Ramin Takloo-Bighash
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19649
UR - https://arxiv.org/abs/2609.19649
AB - Let D be an odd fundamental discriminant, with D=1 permitted, and let r≥ 1 be fixed. We prove that, for every sufficiently large integer ℓ satisfying (−1)^ℓ D>0, the first r traced diagonal Rankin--Cohen brackets Tr_1^|D|[G_ℓ−2e,D,G_ℓ−2e,D]_2e, 1≤ e≤ r, are linearly independent in S_2ℓ(SL_2(ℤ)). Here G_k,D is the Eisenstein series of weight k, level |D|, and nebentypus χ_D. The Petersson formula of Kayath, Lane, Neifeld, Ni, and Xue then implies that at least r normalized Hecke eigenforms f∈ S_2ℓ(SL_2(ℤ)) satisfy L(f⊗χ_D,ℓ)≠ 0. For D=1, this gives, for every fixed r and every sufficiently large K≡ 0pmod 4, at least r level-one eigenforms of weight K with nonzero central value.
ER -