The torsion in the cohomology of wild elliptic fibers

Abstract

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.