The local and global zeta functions of Gauss's curve

Abstract

The singular curve $C\subset \mathbb{P}^2$ defined over $\mathbb{F}_p$ for a prime $p$ by the equation $x^2t^2+y^2t^2+x^2y^2-t^4=0$ is known as Gauss's curve. For $p\equiv 3$ mod $4$, we give a proof that the zeta function of $C$ is \[ Z_{C}(u)=\frac{(1+pu^2)(1+u)^2}{(1-pu)(1-u)}. \] We define the (Hasse-Weil) global zeta function for any geometric genus~1 singular curve and, in particular, find that the global zeta function of $C$ is \[ \zeta_C(s)=\frac{\zeta(s)\zeta(s-1)}{L_E(s)L(s,\chi^{\prime})^{2}}, \] where $E$ is a projective nonsingular model for $C$, $L_E(s)$ is its $L$-function, and $L(s, \chi^{\prime})$ is a Dirichlet $L$-series for a character $\chi^{\prime}$ that we specify. We then consider more generally the ratio $R_{X}(s)$ of the Hasse-Weil global zeta function of a singular curve $X$ and that of its normalization $\widetilde{X}$. We finish with questions about the analytic properties of $R_{X}(s)$.