arXiv subjects

Laura Paladino

Publications and source records attributed to Laura Paladino.

16 recordsLinked to original sources

arXiv2025

On the Hasse principle for divisibility in elliptic curves

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. The validity or the failing of the Hasse principle depends on the elliptic curve $E$ and the field $k$ and, consequently, on the group $\mathrm{Gal}(k(E[p^n])/k)$. For which kind of these groups does the principle hold? For which of them can we find a counterexample? The answer to these questions was known for $n=1,2$, but for $n\geq 3$ they were still open. We show some conditions on the generators of $\mathrm{Gal}(k(E[p^n])/k)$ implying an affirmative answer to the local-global divisibility by $p^n$ in $E$ over $k$, for every $n\geq 2$. We also prove that these conditions are necessary by producing counterexamples in the case when they do not hold. These last results generalize to every power $p^n$, a result obtained by Ranieri for $n=2$.

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arXiv2023

Local-global divisibility on algebraic tori

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 local-global divisibility by any power $p^n$ holds for $T(k)$. We also show that this bound on the dimension is best possible, by providing a counterexample of every dimension $r \geq p-1$. Finally, we prove that under certain hypotheses on the number field generated by the coordinates of the $p^n$-torsion point of $T$, the local-global divisibility still holds for tori of dimension less than $3(p-1)$.

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arXiv2023

A survey of local-global methods for Hilbert's Tenth Problem

Hilbert's Tenth Problem (H10) for a ring R asks for an algorithm to decide correctly, for each $f\in\mathbb{Z}[X_{1},\dots,X_{n}]$, whether the diophantine equation $f(X_{1},...,X_{n})=0$ has a solution in R. The celebrated `Davis-Putnam-Robinson-Matiyasevich theorem' shows that {\bf H10} for $\mathbb{Z}$ is unsolvable, i.e.~there is no such algorithm. Since then, Hilbert's Tenth Problem has been studied in a wide range of rings and fields. Most importantly, for {number fields and in particular for $\mathbb{Q}$}, H10 is still an unsolved problem. Recent work of Eisenträger, Poonen, Koenigsmann, Park, Dittmann, Daans, and others, has dramatically pushed forward what is known in this area, and has made essential use of local-global principles for quadratic forms, and for central simple algebras. We give a concise survey and introduction to this particular rich area of interaction between logic and number theory, without assuming a detailed background of either subject. We also sketch two further directions of future research, one inspired by model theory and one by arithmetic geometry.

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arXiv2023

Testing local-global divisibility at a stable set

We show that the local-global divisibility in commutative algebraic groups defined over number fields can be tested on sets of primes of arbitrary small density, i.e. stable and persistent sets. We also give a new description of the cohomological group giving an obstruction to the problem. In addition, we show new examples of stable sets.

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arXiv2023

Cohomology of groups acting on vector spaces over finite fields

Let ${\mathbf{F}}_q$ be the finite field with $q=p^m$ elements and $G$ be a subgroup of ${\rm{GL}}_n({\mathbf{F}}_q)$. A famous theorem of Nori published in 1987 states that there exists a (non-effective) constant $c(n)$, depending only on $n$, such that if $p>c(n)$ and $G$ acts semisimply on ${\mathbf{F}}_p^n$, then $H^1(G,{\mathbf{F}}_p^n)=0$. We solve the long-standing problem, also considered by Serre of giving an effective proof of Nori's Theorem. Our approach yields the optimal constant $c(n)=n+2$. We also prove a more general version of Nori's theorem, namely, that for all powers $q$ of $p$, if $G$ acts semisimply on ${\mathbf{F}}_q^n$ and $p>n+2$, then $H^1(G,{\mathbf{F}}_q^n)$ is trivial. We apply these results to refine a criterion, proved by Çiperiani and Stix, which gives sufficient conditions for an affirmative answer to a classical question posed by Cassels in the case of abelian varieties over number fields.

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arXiv2022

On 7-division fields of CM elliptic curves

Let $\mathcal{E}$ 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 ${\mathcal{E}}[m]$ the $m$-torsion subgroup of ${\mathcal{E}}$ and by $K_m:=K({\mathcal{E}}[m])$ the $m$-th division field, i.e. the extension of $K$ generated by the coordinates of the points in ${\mathcal{E}}[m]$. We classify all fields $K_7$. In particular we give explicit generators for $K_7/K$ and produce all Galois groups ${\textrm{Gal}}(K_7/K)$. We also show some applications to the Local-Global Divisibility Problem and to modular curves.

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arXiv2022

Local-global questions for divisibility in commutative algebraic groups

This is a survey focusing on the Hasse principle for divisibility of points in commutative algebraic groups and its relation with the Hasse principle for divisibility of elements of the Tate-Shavarevich group in the Weil-Châtelet group. The two local-global subjects arose as a generalization of some classical questions considered respectively by Hasse and Cassels. We describe the deep connection between the two problems and give an overview of the long-established results and the ones achieved during the last twenty years, when the questions were taken up again in a more general setting. In particular, by connecting various results about the two problems, we describe how some recent developments in the first of the two local-global questions imply an answer to Cassel's question, which improves all the results published before about that problem. This answer is best possible over $\mathbb{Q}$. We also describe some links with other similar questions, as for examples the Support Problem and the local-global principle for existence of isogenies of prime degree in elliptic curves.

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arXiv2021

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 $m$-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 prime and $r$ a positive integer, we prove $K(\mathcal{E}[p^r])=K(x_1,ζ_p,y_2)$, where $\{(x_1, y_1),(x_2,y_2)\}$ is a generating system of ${\mathcal{E}}[p^r]$ and $ζ_p$ is a primitive $p$-th root of the unity. If $\mathcal{E}$ has a $K$-rational point of order $p$, then $K(\mathcal{E}[p^r])=K(ζ_{p^r},\sqrt[m_1]{a})$, with $a\in K(ζ_{p^r})$ and $m_1|p^r$. In addition, when $K$ is a number field, we produce an upper bound to the logarithmic height of the discriminant of the extension $K(\mathcal{E}[m])/K$, for all $m\geq 3$. As a consequence, we give an explicit effective version of the hypotheses of the local-global divisibility problem in elliptic curves over number fields.

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arXiv2019

Divisibility questions in commutative algebraic groups

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

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arXiv2018

On 5-torsion of CM elliptic curves

Let $\mathcal{E}$ be an elliptic curve defined over a number field $K$. Let $m$ be a positive integer. We denote by ${\mathcal{E}}[m]$ the $m$-torsion subgroup of $\mathcal{E}$ and by $K_m:=K({\mathcal{E}}[m])$ the number field obtained by adding to $K$ the coordinates of the points of ${\mathcal{E}}[m]$. We describe the fields $K_5$, when $\mathcal{E}$ is a CM elliptic curve defined over $K$, with Weiestrass form either $y^2=x^3+bx$ or $y^2=x^3+c$. In particular we classify the fields $K_5$ in terms of generators, degrees and Galois groups. Furthermore we show some applications of those results to the Local-Global Divisibility Problem, to modular curves and to Shimura curves.

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arXiv2016

On preperiodic points of rational functions defined over $\mathbb{F}_p(t)$

Let $P\in\mathbb{P}_1(\mathbb{Q})$ be a periodic point for a monic polynomial with coefficients in $\mathbb{Z}$. With elementary techniques one sees that the minimal periodicity of $P$ is at most $2$. Recently we proved a generalization of this fact to the set of all rational functions defined over ${\mathbb{Q}}$ with good reduction everywhere (i.e. at any finite place of $\mathbb{Q}$). The set of monic polynomials with coefficients in $\mathbb{Z}$ can be characterized, up to conjugation by elements in PGL$_2({\mathbb{Z}})$, as the set of all rational functions defined over $\mathbb{Q}$ with a totally ramified fixed point in $\mathbb{Q}$ and with good reduction everywhere. Let $p$ be a prime number and let ${\mathbb{F}}_p$ be the field with $p$ elements. In the present paper we consider rational functions defined over the rational global function field ${\mathbb{F}}_p(t)$ with good reduction at every finite place. We prove some bounds for the cardinality of orbits in ${\mathbb{F}}_p(t)\cup \{\infty\}$ for periodic and preperiodic points.

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arXiv2015

Preperiodic points for rational functions defined over a global field in terms of good reductions

Let $ϕ$ be an endomorphism of the projective line defined over a global field $K$. We prove a bound for the cardinality of the set of $K$-rational preperiodic points for $ϕ$ in terms of the number of places of bad reduction. The result is completely new in the function fields case and it is an improvement of the number fields case. An important tool is an $S$-unit equation theorem in 2 variables.

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arXiv2015

Fields generated by torsion points of elliptic curves

Let K be a number field and let $\mathcal{E}$ be an elliptic curve defined over $K$. Let $m$ be a positive integer. We denote by $K(\mathcal{E}[m])$ the number fields obtained by adding to $K$ the coordinates of the $m$-torsion points of $\mathcal{E}$. We look for small (sometimes "minimal") set of generators of $K(\mathcal{E}[m])$. For $m=3$ and $m=4$, we describe explicit generators, degree and Galois groups of the extensions $K(\mathcal{E}[m])/K$.

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arXiv2013

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 $\mathbb{Q} (ζ_p + \bar{ζ_p})$. Let $\mathcal{E}$ 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 $p^n$, 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. Nice and deep works give explicit small constants for elliptic curves defined over a number field of degree at most 5 over $\mathbb{Q}.

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arXiv2011

On the minimal set for counterexamples to the local-global principle

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 give an example that shows that the assumptions of our criterion are necessary.

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arXiv2011

On local-global divisibility by $p^2$ in elliptic curves

Let $ p $ be a prime lager than 3. Let $k$ be a number field, which does not contain the subfield of $\mathbb{Q} (ζ_{p^2})$ of degree $p$ over $\mathbb{Q}$. Suppose that $\mathcal{E}$ is an elliptic curve defined over $k$. We prove that the existence of a counterexample to the local-global divisibility by $p^2$ in $\mathcal{E}$, assures the existence of a $k$-rational point of exact order $p$ in $\mathcal{E}$. Using the Merel Theorem, we then shrunk the known set of primes for which there could be a counterexample to the local-global divisibility by $p^2$.

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