Local-global Divisibility Research Articles

Local–global divisibility on algebraic tori

AbstractWe give a complete answer to the local–global divisibility problem for algebraic tori. In particular, we prove that given an odd prime , if is an algebraic torus of dimension defined over a number field , then the local–global divisibility by any power holds for . We also show that this bound on the dimension is best possible, by providing a counterexample for every dimension . Finally, we prove that under certain hypotheses on the number field generated by the coordinates of the ‐torsion points of , the local–global divisibility still holds for tori of dimension less than .

On 7-division fields of CM elliptic curves

Let be a CM elliptic curve defined over a number field K, with Weiestrass form y^3=x^3+bx or y^2=x^3+c. For every positive integer m, we denote by the m-torsion subgroup of and by the m-th division field, i.e. the extension of K generated by the coordinates of the points in . We classify all the fields K_7. In particular we give explicit generators for K_7/K and produce all the Galois groups textrm{Gal}(K_7/K). We also show some applications to the Local–Global Divisibility Problem and to modular curves.

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 [Formula: see text] ([Formula: see text]), we show that the product of the restriction maps for places in a set of places [Formula: see text] is injective if the Dirichlet density of [Formula: see text] is strictly larger than [Formula: see text]. We give applications to the local–global divisibility problem for elliptic curves and the Hasse principle for flexes on plane cubic curves.

On the division fields of an elliptic curve and an effective bound to the hypotheses of the local-global divisibility

We investigate some aspects of the [Formula: see text]-division field [Formula: see text], where [Formula: see text] is an elliptic curve defined over a field [Formula: see text] with [Formula: see text] and [Formula: see text] is a positive integer. When [Formula: see text], with [Formula: see text] a prime and [Formula: see text] a positive integer, we prove [Formula: see text], where [Formula: see text] is a generating system of [Formula: see text] and [Formula: see text] is a primitive [Formula: see text]th root of unity. If [Formula: see text] has a [Formula: see text]-rational point of order [Formula: see text], then [Formula: see text] for some [Formula: see text] and [Formula: see text]. For every number field [Formula: see text], we produce an upper bound to the logarithmic height of the discriminant of the extension [Formula: see text] for all [Formula: see text]. As a consequence, we consider a version of the local-global divisibility problem in elliptic curves over number fields, where the local conditions are known only for finitely many places.

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

Let $k$ be a number field and let ${\mathcal{A}}$ be a ${\rm GL}_2$-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 ${\mathcal{A}}$ over $k$, then there exists a cyclic extension $\widetilde{k}$ of $k$ of degree bounded by a constant depending on $d$ such that ${\mathcal{A}}$ is $\widetilde{k}$-isogenous to a ${\rm GL}_2$-type variety defined over $\widetilde{k}$ that admits a $\widetilde{k}$-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 Ciperiani and Stix and Creutz.

Divisibility questions in commutative algebraic groups

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 ▪. When A is a principally polarized abelian variety, those conditions imply that the elements of the Tate-Shafarevich group ▪ are divisible by p in the Weil-Châtelet group H1(k,A) and the local-global principle for divisibility by p holds in Hr(k,A), for all r≥0.

On the local-global divisibility over abelian varieties

Let $p \geq 2$ be a prime number and let $k$ be a number field. Let $\mathcal{A}$ be an abelian variety defined over $k$. We prove that if ${\rm Gal} ( k ( {\mathcal{A}}[p] ) / k )$ contains an element $g$ of order dividing $p-1$ not fixing any non-trivial element of ${\mathcal{A}}[p]$ and $H^1 ( {\rm Gal} ( k ( {\mathcal{A}}[p] ) / k ), {\mathcal{A}}[p] )$ is trivial, then the local-global divisibility by $p^n$ holds for ${\mathcal{A}} ( k )$ for every $n \in \mathbb{N}$. Moreover, we prove a similar result without the hypothesis on the triviality of $H^1 ( {\rm Gal} ( k ( {\mathcal{A}}[p] ) / k ) , {\mathcal{A}}[p] )$, in the particular case where ${\mathcal{A}}$ is a principally polarized abelian variety. Then, we get a more precise result in the case when ${\mathcal{A}}$ has dimension $2$. Finally we show with a counterexample that the hypothesis over the order of $g$ is necessary. In the Appendix, we explain how our results are related to a question of Cassels on the divisibility of the Tate-Shafarevich group, studied by Ciperani and Stix.

Counterexamples to the local–global divisibility over elliptic curves

Let $$p \ge 5$$ be a prime number. We find all the possible subgroups G of $$\mathrm{GL}_2 ({\mathbb Z}/ p {\mathbb Z})$$ such that there exist a number field k and an elliptic curve $${\mathcal {E}}$$ defined over k such that the $$\mathrm{Gal}(k ({\mathcal {E}}[p])/k)$$ -module $${\mathcal {E}}[p]$$ is isomorphic to the G-module $$({\mathbb Z}/ p {\mathbb Z})^2$$ and there exists $$n \in {\mathbb N}$$ such that the local–global divisibility by $$p^n$$ does not hold over $${\mathcal {E}}(k)$$ .

Local-global principles for Weil–Châtelet divisibility in positive characteristic

AbstractWe extend existing results characterizing Weil-Châtelet divisibility of locally trivial torsors over number fields to global fields of positive characteristic. Building on work of González-Avilés and Tan, we characterize when local-global divisibility holds in such contexts, providing examples showing that these results are optimal. We give an example of an elliptic curve over a global field of characteristic 2 containing a rational point which is locally divisible by 8, but is not divisible by 8 as well as examples showing that the analogous local-global principle for divisibility in the Weil-Châtelet group can also fail.

On the local–global divisibility of torsion points on elliptic curves and GL2-type varieties

Let p be a prime number and let k be a number field. Let E be an elliptic curve defined over k. We prove that if p is odd, then the local–global divisibility by any power of p holds for the torsion points of E. We also show with an example that the hypothesis over p is necessary.We get a weak generalization of the result on elliptic curves to the larger family of GL2-type varieties over k. In the special case of the abelian surfaces A/k with quaternionic multiplication over k we obtain that for all prime numbers p, except a finite number depending only on the isomorphism class of the ring Endk(A), the local–global divisibility by any power of p holds for the torsion points of A.

On local-global divisibility by p n in elliptic curves

Let p be a prime number and let k be a number field, which does not contain the field ℚ ( ζ p + ζ p ¯ ) . Let ℰ be an elliptic curve defined over k. We prove that if there are no k-rational torsion points of exact order p on ℰ, then the local–global principle holds for divisibility by pn, with n a natural number. As a consequence of the deep theorem of Merel, for p larger than a constant depending only on the degree of k, there are no counterexamples to the local–global divisibility principle. Explicit small constants for elliptic curves defined over a number field of degree at most 5 over ℚ are provided in several deep works of different authors.

On counterexamples to local-global divisibility in commutative algebraic groups

Let k be a number field, let A be a commutative algebraic group defined over k and let n be a positive integer. We will prove that for almost all primes p ∈ k, the existence of a counterexample to the localglobal divisibility by p in A assures the existence of a counterexample to the local-global divisibility by p in A, for all positive integers s. In particular, we will prove the existence of counterexamples to the localglobal divisibility by 2 and 3, for every n ≥ 2, in the case when k = Q and A is an elliptic curve. Furthermore, we will show a method to find numerical examples. We will use this method to produce numerical counterexamples to the local-global divisibility by 2, for every n ≥ 2, in elliptic curves defined over Q.

Elliptic curves with {\mathbb{Q}}({\mathcal{E}}[3])= {\mathbb{Q}}(\zeta _3) and counterexamples to local-global divisibility by 9

Nous donnons une famille F h,β de courbes elliptiques, dependant de deux parametres rationnels non nuls β et h, telle que nous avons la propriete suivante: soit e une courbe elliptique et soit e[3] son sous-groupe de 3-torsion. On a que ℚ(e[3]) = ℚ(ζ 3 ) si et seulement si e est une courbe de la famille F h,β . De plus, nous considerons le probleme de la divisibilite locale-globale par 9 pour les points d'une courbe elliptique. Le nombre 9 est une des rares puissances d'un nombre premier pour laquelle on ne connait pas la reponse a la divisibilite locale-globale dans le cas de tels groupes algebriques. Dans ce papier nous donnons une reponse negative. Nous exhibons des courbes de la famille F h,β , avec des points qui sont localement divisibles par 9 presque partout, mais qui ne sont pas globalement divisibles par 9, sur un corps de nombres de degre au plus 2 sur ℚ(ζ 3 ).

Local–global divisibility by 4 in elliptic curves defined over $${\mathbb{Q}}$$

Let \({\mathcal{E}}\) be an elliptic curve defined over \({\mathbb{Q}}\) . Let \({P\in {\mathcal{E}}(\mathbb {Q})}\) and let q be a positive integer. Assume that for almost all valuations \({v\in \mathbb{Q}}\) , there exist points \({D_v\in {\mathcal{E}}(\mathbb {Q}_v)}\) such that P = qDv. Is it possible to conclude that there exists a point \({D\in {\mathcal{E}}(\mathbb {Q})}\) such that P = qD? A full answer to this question is known when q is a power of almost all primes \({p\in \mathbb{N}}\) , but some cases remain open when \({p\in S=\{2,3,5,7,11,13,17,19,37,43,67,163\}}\) . We now give a complete answer in the case when q = 4.

Cohomology of integer matrices and local-global divisibility on the torus

Soient p ≠ 2 un nombre premier et G un p-groupe de matrices dans SL n (Z), pour un nombre entier n. Dans cet article nous montrons que, pour n < 3(p-1), un certain sous-groupe du groupe de cohomologie H 1 (G, F n p ) est trivial. Nous montrons aussi que cette affirmation peut etre fausse pour n ≥ 3(p - 1). Avec un resultat de Dvornicich et Zannier (voir [2]), nous obtenons que le principe local-global de divisibilite pour p vaut pour tout tore algebrique de dimension n < 3(p - 1).

Local-global divisibility of rational points in some commutative algebraic groups

Let 𝒜 be a commutative algebraic group defined over a number field k. We consider the following question: Let r be a positive integer and let P∈𝒜(k). Suppose that for all but a finite number of primes v of k, we have P=rD v for some D v ∈𝒜(k v ). Can one conclude that there exists D∈𝒜(k) such that P=rD? A complete answer for the case of the multiplicative group 𝔾 m is classical. We study other instances and in particular obtain an affirmative answer when r is a prime and 𝒜 is either an elliptic curve or a torus of small dimension with respect to r. Without restriction on the dimension of a torus, we produce an example showing that the answer can be negative even when r is a prime.