Weil-Chatelet Groups of Rational Elliptic Surfaces

TL;DR

This paper classifies certain rational elliptic surfaces with Galois covers based on an $ ext{L}$-stability condition, and uses Mordell-Weil lattice theory to analyze Weil-Châtelet groups and their restriction maps.

Contribution

It introduces the concept of $ ext{L}$-stability for pairs of rational elliptic surfaces and Galois covers, and applies Mordell-Weil lattice theory to compute kernels of Weil-Châtelet group restriction maps.

Findings

Classification of $ ext{L}$-stable pairs $(S, extgamma)$

Computation of kernels of Weil-Châtelet group restriction maps

Results on injectivity of restriction maps for non-$ ext{L}$-stable pairs

Abstract

We classify pairs $(S, \gamma)$, consisting of a rational elliptic surface $S$ and a Galois cover $\gamma$ of the base, which satisfy a condition we call $\mathcal{L}$-stability. We explain how to use the theory of Mordell-Weil lattices to compute the kernel of the restriction maps of Weil-Chatelet groups for $\mathcal{L}$-stable pairs. We also prove results about the injectivity of restriction maps of Weil-Chatelet groups for some pairs which are not $\mathcal{L}$-stable.