定向Singer圆锥上的特征和与显式Ramanujan二重覆盖

中文摘要 AI 辅助

设$q$为奇数。$\n\mathbb{F}_{q^{3}}$中的迹圆锥确定了$\mathbb{F}_{q^{3}}^{\times} /\mathbb{F}_{q}^{\times}$中的一个Singer差集，以及到$\mathbb{F}_{q^{3}}^{\times} /\mathbb{F}_{q}^{\times 2}$的自然平方类提升。我们研究该提升的奇乘性Fourier系数，并证明其绝对值以$2\sqrt{q}$为界。该证明通过利用相关秩一局部系统的射影Klein四元对称性，改进了朴素的六点Weil界。由此得到的非平凡上循环在其四维上同调上产生一个四元数作用，而Frobenius对称性将相关迹约化为两个Weil尺度的迹。作为应用，定向圆锥给出了$PG(2,q)$的点线关联图的一个显式Singer不变符号。相应的二面体Cayley图是一个连通的Ramanujan二重覆盖。因此，有限几何构造中已知的圆锥提升具有由其奇乘性特征和控制的额外Ramanujan谱性质。

英文摘要

Let $q$ be odd. The trace conic in $\mathbb{F}_{q^3}$ determines a Singer difference set in $\mathbb{F}_{q^3}^\times / \mathbb{F}_q^\times$ and a natural square-class lift to $\mathbb{F}_{q^3}^\times / \mathbb{F}_q^{\times 2}$. We study the odd multiplicative Fourier coefficients of this lift and prove that they are bounded in absolute value by $2\sqrt{q}$. 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 $\mathrm{PG}(2, q)$. 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.

发表机构

• Soochow University（东吴大学）
• National Yang Ming Chiao Tung University（国立阳明交通大学）

机构由 AI 辅助整理，请以论文原文为准。