Fabrizio Barroero

activity

20112026

most citedCounting algebraic integers of fixed degree and bounded height

16 citations · 58 across the 14 of their papers we have counts for

collaborators

18 papers

math.NT2026

Intersecting a curve in an abelian variety with multiples of another curve

Fabrizio Barroero, Gabriel A. Dill, Lars Kuehne

Levin asked what can be said about the locus of points lying on a given curve in $G_{m}^{n}$ that have a non-zero integer multiple on another given curve in $G_{m}^{n}$ f…

math.NT2026

Degeneracy loci and unlikely intersections in abelian schemes

Fabrizio Barroero, Laura Capuano, Tangli Ge +1

Let $A \to S$ be an abelian scheme over a normal quasi-projective variety over a number field, and let $X \subseteq A$ be a positive-dimensional subvarie…

math.NT2024

Hecke orbits and the Mordell-Lang conjecture in distinguished categories

Fabrizio Barroero, Gabriel Andreas Dill

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 varieti…

math.NT2022★ 1 cited

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

Fabrizio Barroero, Laura Capuano, Amos Turchet

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 th…

math.NT2021

Betti maps, Pell equation in polynomials and almost Belyi maps

Fabrizio Barroero, Laura Capuano, Umberto Zannier

We study the Betti map of a particular (but relevant) section of the family of Jacobians of hyperelliptic curves using the polynomial Pell equation $A^{2} - D B^{2} = 1$ , with $A,B,D\in \ma…

math.NT2021

Unlikely Intersections of Curves with Algebraic Subgroups in Semiabelian Varieties

Fabrizio Barroero, Lars Kühne, Harry Schmidt

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 th…

math.NT2021★ 2 cited

Distinguished categories and the Zilber-Pink conjecture

Fabrizio Barroero, Gabriel Andreas Dill

We propose an axiomatic approach towards studying unlikely intersections by introducing the framework of distinguished categories. This includes commutative algebraic groups and mi…

math.NT2020

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

Fabrizio Barroero, Laura Capuano, László Mérai +2

For positive integers $K$ and $L$ , we introduce and study the notion of $K$ -multiplicative dependence over the algebraic closure $F_{p}$ of a finite prime field $…

math.NT2019★ 1 cited

On the Zilber-Pink conjecture for complex abelian varieties

Fabrizio Barroero, Gabriel Andreas Dill

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…

math.NT2019

Torsion points with multiplicatively dependent coordinates on elliptic curves

Fabrizio Barroero, Min Sha

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 th…

math.NT2018★ 8 cited

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

Fabrizio Barroero, Laura Capuano

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…

math.NT2016★ 8 cited

CM relations in fibered powers of elliptic families

Fabrizio Barroero

Let $E_{λ}$ be the Legendre family of elliptic curves. Given $n$ linearly independent points $P_{1}, \dots, P_{n} \in E_{λ} (Q (λ))$ we prove that there are at…

math.NT2016

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

Fabrizio Barroero, Laura Capuano

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 coordina…

math.NT2014★ 15 cited

Linear relations in families of powers of elliptic curves

Fabrizio Barroero, Laura Capuano

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 indepen…

math.NT2013

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

Fabrizio Barroero

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…

math.NT2013★ 16 cited

Counting algebraic integers of fixed degree and bounded height

Fabrizio Barroero

Let $k$ be a number field. For $H \to \infty$ , we give an asymptotic formula for the number of algebraic integers of absolute Weil height bounded by $H$ …

math.NT2012

Counting lattice points and o-minimal structures

Fabrizio Barroero, Martin Widmer

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…

math.NT2011★ 7 cited

Additive unit representations in global fields - A survey

Fabrizio Barroero, Christopher Frei, Robert F. Tichy

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 ho…

Intersecting a curve in an abelian variety with multiples of another curve

Degeneracy loci and unlikely intersections in abelian schemes

Hecke orbits and the Mordell-Lang conjecture in distinguished categories

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

Betti maps, Pell equation in polynomials and almost Belyi maps

Unlikely Intersections of Curves with Algebraic Subgroups in Semiabelian Varieties

Distinguished categories and the Zilber-Pink conjecture

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

On the Zilber-Pink conjecture for complex abelian varieties

Torsion points with multiplicatively dependent coordinates on elliptic curves

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

CM relations in fibered powers of elliptic families

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

Linear relations in families of powers of elliptic curves

Algebraic S-integers of fixed degree and bounded height

Counting algebraic integers of fixed degree and bounded height

Counting lattice points and o-minimal structures

Additive unit representations in global fields - A survey

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