The Hasse principle for finite Galois modules allowing exceptional sets of positive density

TL;DR

This paper investigates a generalized Hasse principle for finite Galois modules with positive density exceptional sets, establishing conditions for injectivity of restriction maps and applying results to elliptic curves and cubic curves.

Contribution

It introduces a new variant of the Hasse principle allowing positive density exceptional sets and proves injectivity conditions for Galois modules with small rank.

Findings

Injectivity of restriction maps when density exceeds a specific threshold

Applications to local-global divisibility on elliptic curves

Hasse principle results for flexes on plane cubic curves

Abstract

We study a variant of the Hasse principle for finite Galois modules, allowing exceptional sets of positive density. For a Galois module whose underlying abelian group is isomorphic to $\mathbb{F}_p^{\oplus r}$ ($r \leq 2$), we show that the product of the restriction maps for places in a set of places $S$ is injective if the Dirichlet density of $S$ is strictly larger than $1 - p^{-r}$. We give applications to the local-global divisibility problem for elliptic curves and the Hasse principle for flexes on plane cubic curves.