Locally trivial torsors that are not Weil-Châtelet divisible

Abstract

Abstract. For every prime p we give infinitely many examples of torsors under abelian varieties over Q that are locally trivial but not divisible by p in the Weil-Châtelet group. We also give an example of a locally trivial torsor under an elliptic curve over Q which is not divisible by 4 in the Weil-Châtelet group. This gives a negative answer to a question of Cassels.

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

“…Cassels' question is considered again in [Baš72], and recently by Ç iperiani and Stix [Ç S13] who showed that, for elliptic curves over Q, the local-global principle for divisibility by p n holds for all prime powers with p ≥ 11. An example showing that it does not hold in general over Q for any p n = 2 n with n ≥ 2 was constructed in [Cre13]. In this note we produce examples settling these questions for the remaining undecided powers of the primes 2 and 3.…”

“…Rather, the elements of X 1 (Q, E) which are proven not to be divisible by 9 in H 1 (Q, E) are those that are not orthogonal to C with respect to the Cassels-Tate pairing. See [Cre13,Theorem 4].…”

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]).…”

Most binary forms come from a pencil of quadrics

“…is a discriminant form locally, but not over Q (see [Cre13,Theorem 11]). Of course, the forms appearing in (1.1) are not generic.…”