The Hasse principle for finite Galois modules allowing exceptional sets of positive density · arXivDesk
1807.09909Jul 26, 201821 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
The Hasse principle for finite Galois modules allowing exceptional sets of positive density
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 Fp⊕r (r≤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
Nearby in the stack
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.