Elliptic-elliptic surfaces and the Hesse pencil

We construct a family of elliptic surfaces with $p_g=q=1$ that arise from base change of the Hesse pencil. We identify explicitly a component of the higher Noether-Lefschetz locus with positive Mordell-Weil rank, and a particular surface having maximal Picard number and defined over $\mathbb Q$. These examples satisfy the infinitesimal Torelli theorem, providing a second proof of the dominance of period map, which was first obtained by Engel-Greer-Ward. A third proof is provided using the Shioda modular surface associated with $\Gamma_0(11)$. Finally, we find birational models for the degenerations at the boundary of the one-dimensional Noether-Lefschetz locus, and extend the period map at those limit points.