On the local-global principle for divisibility in the cohomology of elliptic curves

Abstract

Abstract. For every prime power p n with p = 2 or 3 and n ≥ 2 we give an example of an elliptic curve over Q containing a rational point which is locally divisible by p n but is not divisible by p n . For these same prime powers we construct examples showing that the analogous local-global principle for divisibility in the Weil-Châtelet group can also fail.

Vanishing of Some Galois Cohomology Groups for Elliptic Curves

“…It is our pleasure to thank Jean Gillibert and John Coates for interesting comments and suggestions. We are also grateful to Brendan Creutz for pointing us to [7]. We have also done numerical calculation of the kernel in (7) for other primes.…”

“…We include here a new counter-example for m = 9; the method is quite different from [7] where a first such example was found.…”

Divisibility questions in commutative algebraic groups

“…In particular Cassels questioned if the elements of the Tate-Shafarevich group X(k, E) were divisible by p l in the Weil-Châtelet group H 1 (k, E), for all l. Tate produced soon an affirmative answer for divisibility by p (see [6]). the answer is negative over Q by [12].…”

“…In the case when A is an abelian variety, with dual A ∨ , the triviality of X(k, A[p] ∨ ) implies X(k, A) ⊆ pH r (k, A), for every positive integer r (see [12,Theorem 2.1]). When…”

Counterexamples to the local–global divisibility over elliptic curves

“…Many mathematicians got criterions for the validity of the local-global divisibility principle for many families of commutative algebraic groups, as algebraic tori ( [DZ1] and [Ill]), elliptic curves ( [Cre1], [Cre2], [DZ1], [DZ2], [DZ3], [GR1], [Pal1], [Pal2], [PRV1], [PRV2]), and very recently polarized abelian surfaces ( [GR2]) and GL 2 -type varieties ( [GR3]).…”