The arithmeticity of a Kodaira Fibration is determined by its universal   cover
  Let $S \to C$ be a Kodaira fibration. Here we show that whether or not the algebraic surface $S$ is defined over a number field depends only on the biholomorphic class of its universal cover. 
Gabino Gonz\'alez-Diez and Sebasti\'an Reyes-Carocca
math.AG