On local-global divisibility over ${\rm GL}_2$-type varieties

Abstract

Let k be a number field and let A be a GL 2 -type variety defined over k of dimension d. We show that for every prime number p satisfying certain conditions (see Theorem 2), if the local-global divisibility principle by a power of p does not hold for A over k, then there exists a cyclic extension k of k of degree bounded by a constant depending on d such that A is k-isogenous to a GL 2 -type variety defined over k that admits a k-rational point of order p.Moreover, we explain how our result is related to a question of Cassels on the divisibility of the Tate-Shafarevich group, studied by Proposition 3. Suppose that A is a GL 2 -type variety defined over a number field k and let E be a totally real field that embeds to End k (A) ⊗ Q. Moreover suppose that End k (A) = End k (A). Then det(ρ P ) = χ p , where χ p : Gal(k/k) → Z * p is the p-adic cyclotomic character.

Local-global questions for divisibility in commutative algebraic groups

“…The same authors studied the local-global divisibility especially in the case of abelian varieties of GL 2 -type [46,48].…”

Local-global divisibility on algebraic tori

“…In addition, for elliptic curves an effective version of the hypotheses of Problem 1.1 is produced in [6]. 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.…”

Local–global divisibility on algebraic tori

“…In addition, for elliptic curves, an effective version of the hypotheses of Problem 1.1 is produced in [7]. For principally polarized abelian surfaces in [12], Gillibert and Ranieri proved sufficient conditions for the local–global divisibility by any prime power $$p^n$$, while in [13], they generalized these conditions in order to answer the case of $${\rm GL}_2$$‐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.…”