  We analyze Higgs bundles $(V,\phi)$ on a class of elliptic surfaces $\pi:X\to
B$, whose underlying vector bundle $V$ has vertical determinant and is
fiberwise semistable. We prove that if the spectral curve of $V$ is reduced,
then $\phi$ is vertical, while if $V$ is fiberwise regular with reduced (resp.
integral) spectral curve, and if its rank and second Chern number satisfy an
inequality involving the genus of $B$ and the degree of the fundamental line
bundle of $\pi$ (resp., if the fundamental line bundle is sufficiently ample),
then $\phi$ is scalar. We apply these results to the problem of characterizing
slope-semistable Higgs bundles with vanishing discriminant in terms of the
semistability of their pull-backs via maps from arbitrary (smooth, irreducible,
complete) curves to $X$.
