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Character sums on an oriented singer conic and explicit Ramanujan double covers

  • Pin-Chi Hung
  • Ming-Hsuan Kang
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Abstract

Let q be odd. The trace conic in 𝔽q3 determines a Singer difference set in 𝔽q3Γ— / 𝔽qΓ— and a natural square-class lift to 𝔽q3Γ— / 𝔽qΓ— 2. We study the odd multiplicative Fourier coefficients of this lift and prove that they are bounded in absolute value by 2√q. The proof improves the naive six-puncture Weil bound by exploiting a projective Klein-four symmetry of the associated rank-one local system. The resulting nontrivial cocycle produces a quaternionic action on its four-dimensional cohomology, while Frobenius symmetry reduces the relevant trace to two Weil-scale eigenvalues. As an application, the oriented conic yields an explicit Singer-invariant signing of the point-line incidence graph of PG(2, q). The corresponding dihedral Cayley graph is a connected Ramanujan double cover. Thus a conic lift already known in finite-geometric constructions has an additional Ramanujan spectral property governed by its odd multiplicative character sums.

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Cite

BibTeXRIS
BibTeX
@misc{hung2026character,
  title = {{Character sums on an oriented singer conic and explicit Ramanujan double covers}},
  author = {Pin-Chi Hung and Ming-Hsuan Kang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14332},
  eprint = {2609.14332},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14332},
}
RIS
TY  - UNPB
TI  - Character sums on an oriented singer conic and explicit Ramanujan double covers
AU  - Pin-Chi Hung
AU  - Ming-Hsuan Kang
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.14332
UR  - https://arxiv.org/abs/2609.14332
AB  - Let q be odd. The trace conic in 𝔽_q^3 determines a Singer difference set in 𝔽_q^3^Γ— / 𝔽_q^Γ— and a natural square-class lift to 𝔽_q^3^Γ— / 𝔽_q^Γ— 2. We study the odd multiplicative Fourier coefficients of this lift and prove that they are bounded in absolute value by 2√q. The proof improves the naive six-puncture Weil bound by exploiting a projective Klein-four symmetry of the associated rank-one local system. The resulting nontrivial cocycle produces a quaternionic action on its four-dimensional cohomology, while Frobenius symmetry reduces the relevant trace to two Weil-scale eigenvalues. As an application, the oriented conic yields an explicit Singer-invariant signing of the point-line incidence graph of PG(2, q). The corresponding dihedral Cayley graph is a connected Ramanujan double cover. Thus a conic lift already known in finite-geometric constructions has an additional Ramanujan spectral property governed by its odd multiplicative character sums.
ER  -