Toric Elliptic Pairs with Picard Number Three

An elliptic pair $(X,C)$ is a generalization of a rational elliptic fibration $X\to {\mathbb{P}}^1$ with fiber $C,$ introduced in \cite{jenia_blowup}. Here, $X$ is a projective rational surface with log terminal singularities, and $C$ is an irreducible curve contained in the smooth locus of $X,$ with $p_a(C)=1$ and $C^2=0.$ These naturally arise as blowups $X:={\text{Bl}}_e({\mathbb{P}}_{\mathrm{\Delta }})$ of projective toric surfaces, whose Newton polygon is elliptic. The order of $\mathcal{O}(C){\mathrm{\mid }}_C$ in ${\text{Pic}}^0(C)$ gives a quantitative way to check if $X$ is an elliptic fibration, which is equivalent to finiteness of the order. We call $\mathrm{\Delta }$ a Lang-Trotter polygon when this order is infinite, in which case $\overline{\text{Eff}({\text{Bl}}_e({\mathbb{P}}_{\mathrm{\Delta }}))}$ is non-polyhedral. The paper \cite{lizzie} shows there are exactly $3$ elliptic triangles up to ${\text{SL}}_2(\mathbb{Z}),$ none of which is Lang-Trotter. The paper \cite{jenia_blowup} gives an infinite family of Lang-Trotter pentagons and heptagons, and various examples of other polygons when $\rho ({\mathbb{P}}_{\mathrm{\Delta }})>2.$ Remark 4.7 in the paper asks if any Lang-Trotter quadrilaterals exist, and we answer this in the negative by studying the curves in the Zariski Decomposition of $K_X+C\mathrm{.}$