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

Abstract

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.