Arithmetic Proportional Elliptic Configurations with Comparatively Large Number of Irreducible Components

Abstract

Let $T$ be an arithmetic proportional elliptic configuration on a bi-elliptic surface $A \sqrt{-d}$ with complex multiplication by an imaginary quadratic number field $\mathbb{Q}(\sqrt{-d})$. The present note establishes that if $T$ has $s$ singular points and \[ 4s − 5 \leq h \leq 4s \] irreducible smooth elliptic components, then $d = 3$ and $T$ is $\mathrm{Aut}(A \sqrt{-3}−$ equivalent to Hirzebruch’s example $T^{(1,4)}_{\sqrt{-3}}$ with a unique singular point and 4 irreducible components.