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Fabrizio Barroero

Publications and source records attributed to Fabrizio Barroero.

16 recordsLinked to original sources
arXiv2025

Hecke orbits and the Mordell-Lang conjecture in distinguished categories

Inspired by recent work of Aslanyan and Daw, we introduce the notion of $Σ$-orbits in the general framework of distinguished categories. In the setting of connected Shimura varieties, this concept contains many instances of (generalized) Hecke orbits from the literature. In the setting of semiabelian varieties, a $Σ$-orbit is a subgroup of finite rank. We show that our $Σ$-orbits have useful functorial properties and we use them to formulate two general statements of Mordell-Lang type (one of them implying the other one). We prove an analogue of a recent theorem of Aslanyan and Daw in this general setting, which we apply to deduce an unconditional result about unlikely intersections in a fibered power of the Legendre family. In an appendix, we prove an unconditional Zilber-Pink result for subvarieties of $\mathcal{A}_g$ that cannot be defined over the algebraic numbers.

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arXiv2024

Distinguished categories and the Zilber-Pink conjecture

We propose an axiomatic approach towards studying unlikely intersections by introducing the framework of distinguished categories. This includes commutative algebraic groups and mixed Shimura varieties. It allows us to define all basic concepts of the field and prove some fundamental facts about them, e.g. the defect condition. In some categories that we call very distinguished, we are able to show some implications between Zilber-Pink statements with respect to base change. This yields unconditional results, i.e. the Zilber-Pink conjecture for a complex curve in $\mathcal{A}_2$ that cannot be defined over $\bar{\mathbb{Q}}$, a complex curve in the $g$-th fibered power of the Legendre family, and a complex curve in the base change of a semiabelian variety over $\bar{\mathbb{Q}}$.

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arXiv2023

Unlikely Intersections in families of abelian varieties and the polynomial Pell equation

Let S be a smooth irreducible curve defined over a number field k and consider an abelian scheme A over S and a curve C inside A, both defined over k. In previous works, we proved that when A is a fibered product of elliptic schemes, if C is not contained in a proper subgroup scheme of A, then it contains at most finitely many points that belong to a flat subgroup scheme of codimension at least 2. In this article, we continue our investigation and settle the crucial case of powers of simple abelian schemes of relative dimension g bigger or equal than 2. This, combined with the above mentioned result and work by Habegger and Pila, gives the statement for general abelian schemes. These results have applications in the study of solvability of almost-Pell equations in polynomials.

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arXiv2022

Unlikely Intersections of Curves with Algebraic Subgroups in Semiabelian Varieties

Let $G$ be a semiabelian variety and $C$ a curve in $G$ that is not contained in a proper algebraic subgroup of $G$. In this situation, conjectures of Pink and Zilber imply that there are at most finitely many points contained in the so-called unlikely intersections of $C$ with subgroups of codimension at least $2$. In this note, we establish this assertion for general semiabelian varieties over $\bar{\mathbb{Q}}$. This extends results of Maurin and Bombieri, Habegger, Masser, and Zannier in the toric case as well as Habegger and Pila in the abelian case.

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arXiv2022

Greatest Common Divisor results on semiabelian varieties and a Conjecture of Silverman

A divisibility sequence is a sequence of integers $\{d_n\}$ such that $d_m$ divides $d_n$ if $m$ divides $n$. Results of Bugeaud, Corvaja, Zannier, among others, have shown that the gcd of two divisibility sequences corresponding to subgroups of the multiplicative group grows in a controlled way. Silverman conjectured that a similar behaviour should appear in many algebraic groups. We extend results by Ghioca-Hsia-Tucker and Silverman for elliptic curves and prove an analogue of Silverman's conjecture over function fields for abelian and split semiabelian varieties and some generalizations of this result. We employ tools coming from the theory of unlikely intersections as well as properties of the so-called Betti map associated to a section of an abelian scheme.

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arXiv2021

Betti maps, Pell equation in polynomials and almost Belyi maps

We study the Betti map of a particular (but relevant) section of the family of Jacobians of hyperelliptic curves using the polynomial Pell equation $A^2-DB^2=1$, with $A,B,D\in \mathbb C[t]$ and certain ramified covers ${\mathbb P}^1\to {\mathbb P}^1$ arising from such equation and having heavy constrains on their ramification. In particular, we obtain a special case of a result of André, Covaja and Zannier on the submersivity of the Betti map by studying the locus of the polynomials $D$ that fit in a Pell equation inside the space of polynomials of fixed even degree. Moreover, Riemann Existence Theorem associates to the above-mentioned covers certain permutation representations: we are able to characterize the representations corresponding to "primitive" solutions of the Pell equation or to powers of solutions of lower degree and give a combinatorial description of these representations when $D$ has degree 4. In turn, this characterization gives back some precise information about the rational values of the Betti map.

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arXiv2021

Multiplicative and linear dependence in finite fields and on elliptic curves modulo primes

For positive integers $K$ and $L$, we introduce and study the notion of $K$-multiplicative dependence over the algebraic closure $\overline{\mathbb{F}}_p$ of a finite prime field $\mathbb{F}_p$, as well as $L$-linear dependence of points on elliptic curves in reduction modulo primes. One of our main results shows that, given non-zero rational functions $φ_1,\ldots,φ_m, \varrho_1,\ldots,\varrho_n\in\mathbb{Q}(X)$ and an elliptic curve $E$ defined over the integers $\mathbb{Z}$, for any sufficiently large prime $p$, for all but finitely many $α\in\overline{\mathbb{F}}_p$, at most one of the following two can happen: $φ_1(α),\ldots,φ_m(α)$ are $K$-multiplicatively dependent or the points $(\varrho_1(α),\cdot), \ldots,(\varrho_n(α),\cdot)$ are $L$-linearly dependent on the reduction of $E$ modulo $p$. As one of our main tools, we prove a general statement about the intersection of an irreducible curve in the split semiabelian variety $\mathbb{G}_{\mathrm{m}}^m \times E^n$ with the algebraic subgroups of codimension at least $2$. As an application of our results, we improve a result of M. C. Chang and extend a result of J. F. Voloch about elements of large order in finite fields in some special cases.

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arXiv2020

Torsion points with multiplicatively dependent coordinates on elliptic curves

In this paper, we study the finiteness problem of torsion points on an elliptic curve whose coordinates satisfy some multiplicative dependence relations. In particular, we prove that on an elliptic curve defined over a number field there are only finitely many torsion points whose coordinates are multiplicatively dependent. Moreover, we produce an effective result when the elliptic curve is defined over the rational numbers or has complex multiplication.

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arXiv2019

On the Zilber-Pink conjecture for complex abelian varieties

In this article, we prove that the Zilber-Pink conjecture for abelian varieties over an arbitrary field of characteristic $0$ is implied by the same statement for abelian varieties over the algebraic numbers. More precisely, the conjecture holds for subvarieties of dimension at most $m$ in the abelian variety $A$ if it holds for subvarieties of dimension at most $m$ in the largest abelian subvariety of $A$ that is isomorphic to an abelian variety defined over $\bar{ \mathbb{Q}}$.

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arXiv2018

CM relations in fibered powers of elliptic families

Let $E_λ$ be the Legendre family of elliptic curves. Given $n$ linearly independent points $P_1,\dots , P_n \in E_λ\left(\overline{\mathbb{Q}(λ)}\right)$ we prove that there are at most finitely many complex numbers $λ_0$ such that $E_{λ_0} $ has complex multiplication and $P_1(λ_0), \dots ,P_n(λ_0)$ are dependent over $End(E_{λ_0})$. This implies a positive answer to a question of Bertrand and, combined with a previous work in collaboration with Capuano, proves the Zilber-Pink conjecture for a curve in a fibered power of an elliptic scheme when everything is defined over $\overline{\mathbb{Q}}$.

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arXiv2017

Unlikely intersections in products of families of elliptic curves and the multiplicative group

Let $E_λ$ be the Legendre elliptic curve of equation $Y^2=X(X-1)(X-λ)$. We recently proved that, given $n$ linearly independent points $P_1(λ), \dots,P_n(λ)$ on $E_λ$ with coordinates in $\bar{\mathbb{Q}(λ)}$, there are at most finitely many complex numbers $λ_0$ such that the points $P_1(λ_0), \dots,P_n(λ_0)$ satisfy two independent relations on $E_{λ_0}$. In this article we continue our investigations on Unlikely Intersections in families of abelian varieties and consider the case of a curve in a product of two non-isogenous families of elliptic curves and in a family of split semi-abelian varieties.

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arXiv2015

Linear relations in families of powers of elliptic curves

Motivated by recent work of Masser and Zannier on simultaneous torsion on the Legendre elliptic curve $E_λ$ of equation $Y^2=X(X-1)(X-λ)$, we prove that, given $n$ linearly independent points $P_1(λ), ...,P_n(λ)$ on $E_λ$ with coordinates in $\bar{\mathbb{Q}(λ)}$, there are at most finitely many complex numbers $λ_0$ such that the points $P_1(λ_0), ...,P_n(λ_0)$ satisfy two independent relations on $E_{λ_0}$. This is a special case of conjectures about Unlikely Intersections on families of abelian varieties.

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arXiv2014

Algebraic $S$-integers of fixed degree and bounded height

Let $k$ be a number field and $S$ a finite set of places of $k$ containing the archimedean ones. We count the number of algebraic points of bounded height whose coordinates lie in the ring of $S$-integers of $k$. Moreover, we give an asymptotic formula for the number of $\bar{S}$-integers of bounded height and fixed degree over $k$, where $\bar{S}$ is the set of places of $\bar{k}$ lying above the ones in $S$.

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arXiv2013

Counting lattice points and o-minimal structures

Let $Λ$ be a lattice in $\R^n$, and let $Z\subseteq \R^{m+n}$ be a definable family in an o-minimal structure over $\R$. We give sharp estimates for the number of lattice points in the fibers $Z_T={x\in \R^n: (T,x)\in Z}$. Along the way we show that for any subspace $Σ\subseteq\R^n$ of dimension $j>0$ the $j$-volume of the orthogonal projection of $Z_T$ to $Σ$ is, up to a constant depending only on the family $Z$, bounded by the maximal $j$-dimensional volume of the orthogonal projections to the $j$-dimensional coordinate subspaces.

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arXiv2011

Additive unit representations in global fields - A survey

We give an overview on recent results concerning additive unit representations. Furthermore the solutions of some open questions are included. The central problem is whether and how certain rings are (additively) generated by their units. This has been investigated for several types of rings related to global fields, most importantly rings of algebraic integers. We also state some open problems and conjectures which we consider to be important in this field.

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