  Given an elliptic fibration $f \colon X \to S$ over the spectrum of a
complete discrete valuation ring with algebraically closed residue field, we
use a Hochschild--Serre spectral sequence to express the torsion in $R^1f_\ast
\mathscr{O}_X$ as the first group cohomology $H^1(G,H^0(S^\prime,
\mathscr{O}_{S^\prime}))$. Here, $G$ is the Galois group of the maximal
extension $K^\prime / K$ such that the normalization of $X \times_S S^\prime$
induces an \'etale covering of $X$, where $S^\prime$ is the normalization of
$S$ in $K^\prime$. The case where $S$ is a Dedekind scheme is easily reduced to
the local case. Moreover, we generalize to higher-dimensional fibrations.
