The torsion in the cohomology of wild elliptic fibers

Abstract

Given an elliptic fibration $f:X\to S$ 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 $R^1{f}_{\ast }{\mathcal{O}}_X$ as the first group cohomology $H^1(G,H^0(S^{\prime },{\mathcal{O}}_{S^{\prime }}))$. 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 étale 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.

14 pages, final manuscript to appear in Journal of Pure and Applied Algebra, March 2021