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Central values of additive twists of automorphic L-series
Abstract
We study series L(x,s):=∑n ρ(n)e2πi n x/ns, where ρ(n) are Fourier coefficients of an automorphic form ϕ for a discrete cofinite subgroup Γ⊂ SL(2,ℝ) and x∈ℝ. Let b∈ℝ∪{∞} represent a cusp of Γ. Ordering the orbit Γb by increasing denominators, we prove that the values L(x,1/2) for x∈Γb, suitably normalized, become equidistributed according to a standard complex Gaussian law, under natural assumptions on the Laplace eigenvalue of ϕ. Previously this was known either when Γ=SL(2,ℤ) or when ϕ is associated to a holomorphic cusp form. This applies in particular to central values at s=1/2 of additive twists of Maass form L-functions ∑n≥1λ(n)e2πi n x/ns, or to the Dirichlet-theta series ∑n≥1χ(n)e2πi n2 x/ns.
Related
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BibTeX
@misc{bettin2026central,
title = {{Central values of additive twists of automorphic $L$-series}},
author = {Sandro Bettin and Sandro Bettin and Jungwon Lee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.42008},
eprint = {2610.42008},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.42008},
}
RIS
TY - UNPB
TI - Central values of additive twists of automorphic L-series
AU - Sandro Bettin
AU - Sandro Bettin
AU - Jungwon Lee
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.42008
UR - https://arxiv.org/abs/2610.42008
AB - We study series L(x,s):=∑_n ρ(n)e^2πi n x/n^s, where ρ(n) are Fourier coefficients of an automorphic form ϕ for a discrete cofinite subgroup Γ⊂ SL(2,ℝ) and x∈ℝ. Let b∈ℝ∪{∞} represent a cusp of Γ. Ordering the orbit Γb by increasing denominators, we prove that the values L(x,1/2) for x∈Γb, suitably normalized, become equidistributed according to a standard complex Gaussian law, under natural assumptions on the Laplace eigenvalue of ϕ. Previously this was known either when Γ=SL(2,ℤ) or when ϕ is associated to a holomorphic cusp form. This applies in particular to central values at s=1/2 of additive twists of Maass form L-functions ∑_n≥1λ(n)e^2πi n x/n^s, or to the Dirichlet-theta series ∑_n≥1χ(n)e^2πi n^2 x/n^s.
ER -