Divisibility questions in commutative algebraic groups

Abstract

Let k be a number field, let A be a commutative algebraic group defined over k and let p be a prime number. Let A[p] denote the p-torsion subgroup of A. We give some sufficient conditions for the local-global divisibility by p in A and the triviality of the Tate-Shafarevich group X(k, A[p]). When A is a principally polarized abelian variety, those conditions imply that the elements of the Tate-Shafarevich group X(k, A) are divisible by p in the Weil-Châtelet group H 1 (k, A) and the local-global principle for divisibility by p holds in H r (k, A), for all r ≥ 0.

On 7-division fields of CM elliptic curves

“…The first application concerns the following local-global question that was stated in [10] by R. Dvornicich and U. Zannier as a generalization of a particular case of the famous Hasse principle on quadratic forms (for further details one can see [11], [7], [12], [20] and [19] among others; Dvornicich and the corresponding author also produced a survey [8] about this topic). Problem 8.1 (Dvornicich, Zannier, 2001).…”

Cohomology of groups acting on vector spaces over finite fields

“…This last step in particular considers the cohomology of a simple group S with values in F n q and gives a proof of Corollary 1.3. The techniques of proof are similar to those used in [24], where the investigation is about the triviality of a subgroup of H 1 (G, F q ), under stronger hypotheses. Anyway here there is an important improvement in the results of every step and there are also additional steps that are fundamental to get the new conclusion.…”

“…Anyway here there is an important improvement in the results of every step and there are also additional steps that are fundamental to get the new conclusion. The proofs of some of the lemmas in the next subsection can be deduced from the results in [24], but we prefer to state all of them explicitly here, in order to have a more self-contained paper for the reader's convenience.…”

Local-global divisibility on algebraic tori

“…For principally polarized abelian surfaces in [12] the authors proved sufficient conditions for the local-global divisibility by any prime power p l , while in [13] they generalized these conditions in order to answer the case of GL 2 -type varieties (see also [11]). Furthermore, in [17] the third author produced conditions for the local-global p-divisibility for a general commutative algebraic group. In the case of abelian varieties, the problem is also linked to a classical question posed by Cassels in 1962 on the p-divisibility of the Tate-Shafarevich group (see [2,3,5]).…”