  We discuss the birational geometry and the Kodaira dimension of certain
varieties previously constructed by Schreieder, proving that in any dimension
they admit an elliptic fibration and they are not of general type. The
$l$-dimensional variety $Y_n^l$, which is the quotient of the product of a
certain curve $C_n$ by itself $l$ times by a group
$G\simeq\left(\mathbb{Z}/n\mathbb{Z}\right)^{l-1}$ of automorphisms, was
constructed by Schreieder to obtain varieties with prescribed Hodge numbers. If
$n=3^c$ Schreieder constructed an explicit smooth birational model of it, and
Flapan proved: the Kodaira dimension of this smooth model is 1, if $c>1$; if
$l=2$ it is a modular elliptic surface; if $l=3$ it admits a fibration in K3
surfaces.
  In this paper we generalize these results: without any assumption on $n$ and
$l$ we prove that $Y_n^l$ admits many elliptic fibrations and its Kodaira
dimension is at most 1. Moreover, if $l=2$ its minimal resolution is a modular
elliptic surface, obtained by a base change of order $n$ on a specific extremal
rational elliptic surface; if $l\geq 3$ it has a birational model which admits
a fibration in K3 surfaces and a fibration in $(l-1)$-dimensional varieties of
Kodaira dimension at most 0.
