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

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 ${\mathbb{F}}_p^{\oplus r}$ ($r\le 2$), 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.

arXiv comment21 pages; The main theorem is generalized and the proof is revised according to J.-P. Serre's suggestion. This version also contains general results on the Hasse principle for finite Galois modules allowing exceptional sets of positive density; to appear in International Journal of Number Theory