Uniform non-vanishing of prime quadratic twists of standard and symmetric square L-functions
Abstract
Let Sk be the space of cusp forms of weight k for SL2(ℤ), with dim Sk≥1, and let χp=(·/p) for odd primes p. For fixed s with Re(s)>(k+1)/2, we prove that the proportion of primes p≤ X satisfying L(h,χp,s)=0 tends to zero as X→∞, uniformly over nonzero h∈ Sk. We also prove an analogous statement for nontrivial linear combinations of twisted symmetric square L-values of the normalized Hecke eigenforms when Re(s)>k. The main ingredient of the proof is a non-concentration theorem for random Euler products under a pairwise local separation condition. As applications, we obtain spanning and density-one basis results in Sk for the kernels associated with twisted standard and symmetric square L-functions, with explicit basis criteria in specified right half-planes when dim Sk=2 or $3$. We show that almost every tuple of dim Sk primes gives a basis of Sk∗ from twisted periods at every noncentral critical index. We also prove a density-one result for rational bases of Sk obtained from traces of Rankin-Cohen brackets of Eisenstein series at prime levels.
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BibTeX
@misc{ni2026uniform,
title = {{Uniform non-vanishing of prime quadratic twists of standard and symmetric square $L$-functions}},
author = {Tianyu Ni},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11974},
eprint = {2610.11974},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11974},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Uniform non-vanishing of prime quadratic twists of standard and symmetric square L-functions
AU - Tianyu Ni
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11974
UR - https://arxiv.org/abs/2610.11974
AB - Let S_k be the space of cusp forms of weight k for SL_2(ℤ), with dim S_k≥1, and let χ_p=(·/p) for odd primes p. For fixed s with Re(s)>(k+1)/2, we prove that the proportion of primes p≤ X satisfying L(h,χ_p,s)=0 tends to zero as X→∞, uniformly over nonzero h∈ S_k. We also prove an analogous statement for nontrivial linear combinations of twisted symmetric square L-values of the normalized Hecke eigenforms when Re(s)>k. The main ingredient of the proof is a non-concentration theorem for random Euler products under a pairwise local separation condition. As applications, we obtain spanning and density-one basis results in S_k for the kernels associated with twisted standard and symmetric square L-functions, with explicit basis criteria in specified right half-planes when dim S_k=2 or $3$. We show that almost every tuple of dim S_k primes gives a basis of S_k^∗ from twisted periods at every noncentral critical index. We also prove a density-one result for rational bases of S_k obtained from traces of Rankin-Cohen brackets of Eisenstein series at prime levels.
ER -