Related papers: On local-global divisibility by $p^n$ in elliptic …

Let $ p $ be a prime lager than 3. Let $k$ be a number field, which does not contain the subfield of $\mathbb{Q} (\zeta_{p^2})$ of degree $p$ over $\mathbb{Q}$. Suppose that $\mathcal{E}$ is an elliptic curve defined over $k$. We prove that…

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…

We consider the local-global principle for divisibility in the Mordell-Weil group of a CM elliptic curve defined over a number field. For each prime $p$ we give sharp lower bounds on the degree $d$ of a number field over which there exists…

For every prime power p^n with p = 2 or 3 and n > 1 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…

Let $p \geq 5$ be a prime number. We find all the possible subgroups $G$ of ${\rm GL}_2 ( \mathbb{Z} / p \mathbb{Z} )$ such that there exists a number field $k$ and an elliptic curve ${\mathcal{E}}$ defined over $k$ such that the ${\rm Gal}…

Let $p$ be a prime number and $n$ a positive integer. Let $E$ be an elliptic curve defined over a number field $k$. It is known that the local-global divisibility by $p$ holds in $E/k$, but for powers of $p^n$ counterexamples may appear.…

We investigate some aspects of the $m$-division field $K({\mathcal{E}}[m])$, where $\mathcal{E}$ is an elliptic curve defined over a field $K$ with ${\textrm{char}}(K)\neq 2,3$ and $m$ is a positive integer. When $m=p^r$, with $p\geq 5$ a…

We extend existing results characterizing Weil-Ch\^atelet divisibility of locally trivial torsors over number fields to global fields of positive characteristic. Building on work of Gonz\'alez-Avil\'es and Tan, we characterize when…

We study the rigidity of the local conditions in two well-known local-global principles for elliptic curves over number fields. In particular, we consider a local-global principle for torsion due to Serre and Katz, and one for isogenies due…

Let $E$ be an elliptic surface over the curve $C$, defined over a number field $k$, let $P$ be a section of $E$, and let $\ell$ be a rational prime. For any non-singular fibre $E_t$, we bound the number of points $Q$ on $E_t$ of (algebraic)…

Using the rank of the Mordell-Weil group $E(\mathbb{Q})$ of an elliptic curve $E$ over $\mathbb{Q}$, we give a lower bound of the class number of the number field $\mathbb{Q}(E[p^n])$ generated by $p^n$-division points of $E$ when the curve…

We give a complete answer to the local-global divisibility problem for algebraic tori. In particular, we prove that given an odd prime $p$, if $T$ is an algebraic torus of dimension $r< p-1$ defined over a number field $k$, then the…

Let K be a number field. We consider a local-global principle for elliptic curves E/K that admit (or do not admit) a rational isogeny of prime degree n. For suitable K (including K=Q), we prove that this principle holds when n = 1 mod 4,…

Let $E$ be an elliptic curve over a number field $k$ of degree $d$ that admits a $k$-rational isogeny of prime degree $p$. We study the question of finding a uniform bound on $p$ that depends only on $d$, and obtain, under a certain…

We give a simple proof of the well-known divisibility by 2 condition for rational points on elliptic curves with rational 2-torsion. As an application of the explicit division by $2^n$ formulas obtained in Sec.2, we construct versal…

We prove that only for powers of 2 and 3 could occur counterexamples to the local-global divisibility principle for elliptic curves defined over the rationals. For we refine our previous criterion for the validity of the principle. We also…

Let $m$ be a positive integer and let $E$ be an elliptic curve over $\mathbb{Q}$ with the property that $m\mid#E(\mathbb{F}_p)$ for a density $1$ set of primes $p$. Building upon work of Katz and Harron-Snowden, we study the probability…

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…

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…

Let $K$ be a global field and let $E/K$ be an elliptic curve with a $K$-rational point of prime order $p$. In this paper we are interested in how often the (global) Tamagawa number $c(E/K)$ of $E/K$ is divisible by $p$. This is a natural…