The torsion in the cohomology of wild elliptic fibers

TL;DR

This paper investigates the torsion in the first cohomology sheaf of elliptic fibrations over discrete valuation rings, expressing it via Galois cohomology and extending results to higher dimensions.

Contribution

It introduces a spectral sequence approach to describe torsion in cohomology sheaves of elliptic fibrations, generalizing to higher-dimensional cases.

Findings

Expresses torsion as first Galois cohomology group.

Reduces the Dedekind scheme case to local analysis.

Generalizes results to higher-dimensional fibrations.

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.