arXiv subjects

Laura Capuano

Publications and source records attributed to Laura Capuano.

17 recordsLinked to original sources

arXiv2025

If a machine did it, it is probably transcendental (even $p$-adically)

Continued fraction expansions provide a well-established bridge between algebraic properties of numbers and combinatorics on words. In this article, we investigate the algebraicity of $p$-adic numbers whose continued fractions arise from certain classes of words which generalize the classical automatic, periodic and palindromic words. Our main result shows that, under mild conditions on the $p$-adic continued fraction expansion, such numbers are either algebraic of degree at most 2 or transcendental. This result provides an analogue of results of Bugeaud and Adamczewski-Bugeaud in the real setting and extends previous works that were limited to specific choices of $p$-adic floor functions and less general classes of words.

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arXiv2024

p-Adically convergent loci in varieties arising from periodic continued fractions

Inspired by several alternative definitions of continued fraction expansions for elements in $\mathbb Q_p$, we study $p$-adically convergent periodic continued fractions with partial quotients in $\mathbb Z[1/p]$. To this end, following a previous work by Brock, Elkies, and Jordan, we consider certain algebraic varieties whose points represent formal periodic continued fractions with period and preperiod of fixed lengths, satisfying a given quadratic equation. We then focus on the $p$-adically convergent loci of these varieties, characterizing the zero and one-dimensional cases.

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arXiv2023

On $\mathfrak{P}$-adic continued fractions with extraneous denominators: some explicit finiteness results

Let $K$ be a number field. We show that, up to allowing a finite set of denominators in the partial quotients, it is possible to define algorithms for $\mathfrak P$-adic continued fractions satisfying the finiteness property on $K$ for every prime ideal $\mathfrak P$ of sufficiently large norm. This provides, in particular, a new algorithmic approach to the construction of division chains in number fields.

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arXiv2022

Quaternionic $p$-adic continued fractions

We develop a theory of $p$-adic continued fractions for a quaternion algebra $B$ over $\mathbb Q$ ramified at a rational prime $p$. Many properties holding in the commutative case can be proven also in this setting. In particular, we focus our attention on the characterization of elements having a finite continued fraction expansion. By means of a suitable notion of quaternionic height, we prove a criterion for finiteness. Furthermore, we draw some consequences about the solutions of a family of quadratic polynomial equations with coefficients in $B$.

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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 Andr\'e, 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

On the finiteness of $\mathfrak{P}$-adic continued fractions for number fields

For a prime ideal $\mathfrak{P}$ of the ring of integers of a number field $K$, we give a general definition of $\mathfrak{P}$-adic continued fraction, which also includes classical definitions of continued fractions in the field of $p$--adic numbers. We give some necessary and sufficient conditions on $K$ ensuring that every $\alpha\in K$ admits a finite $\mathfrak{P}$-adic continued fraction expansion for all but finitely many $\mathfrak{P}$, addressing a similar problem posed by Rosen in the archimedean setting.

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arXiv2020

On periodicity of $p$-adic Browkin continued fractions

The classical theory of continued fractions has been widely studied for centuries for its important properties of good approximation, and more recently it has been generalized to $p$-adic numbers where it presents many differences with respect to the real case. In this paper we investigate periodicity for the $p$-adic continued fractions introduced by Browkin. We give some necessary and sufficient conditions for periodicity in general, although a full characterization of $p$-adic numbers having purely periodic Browkin continued fraction expansion is still missing. In the second part of the paper, we describe a general procedure to construct square roots of integers having periodic Browkin $p$-adic continued fraction expansion of prescribed even period length. As a consequence, we prove that, for every $n \ge 1$, there exist infinitely many $\sqrt{m}\in \QQ_p$ with periodic Browkin expansion of period $2^n$, extending a previous result of Bedocchi obtained for $n=1$.

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arXiv2020

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 $\varphi_1,\ldots,\varphi_m, \varrho_1,\ldots,\varrho_n\in\mathbb{Q}(X)$ and an elliptic curve $E$ defined over the integers $\mathbb{Z}$, for any sufficiently large prime $p$, for all but finitely many $\alpha\in\overline{\mathbb{F}}_p$, at most one of the following two can happen: $\varphi_1(\alpha),\ldots,\varphi_m(\alpha)$ are $K$-multiplicatively dependent or the points $(\varrho_1(\alpha),\cdot), \ldots,(\varrho_n(\alpha),\cdot)$ 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

A note on cyclotomic polynomials and Linear Feedback Shift Registers

Linear Feedback Shift Registers (LFRS) are tools commonly used in cryptography in many different context, for example as pseudo-random numbers generators. In this paper we characterize LFRS with certain symmetry properties. Related to this question we also classify polynomials f of degree n satisfying the property that if a is a root of f then $f(a^n)=0$. The classification heavily depends on the choice of the fields of coefficients of the polynomial; we consider the cases $K=\mathbb{F}_p$ and $K=\mathbb{Q}$.

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arXiv2019

Lang-Vojta Conjecture over function fields for surfaces dominating $\mathbb{G}_m^2$

We prove the nonsplit case of the Lang-Vojta conjecture over function fields for surfaces of log general type that are ramified covers of $\mathbb{G}_m^2$. This extends results of Corvaja and Zannier, who proved the conjecture in the split case, and results of Corvaja and Zannier and the second author that were obtained in the case of the complement of a degree four and three component divisor in $\mathbb{P}^2$. We follow the strategy developed by Corvaja and Zannier and make explicit all the constants involved.

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arXiv2018

An effective criterion for periodicity of l-adic continued fractions

The theory of continued fractions has been generalized to l-adic numbers by several authors and presents many differences with respect to the real case. In the present paper we investigate the expansion of rationals and quadratic irrationals for the l-adic continued fractions introduced by Ruban. In this case, rational numbers may have a periodic non-terminating continued fraction expansion, moreover, for quadratic irrational numbers, no analogue of Lagrange's theorem holds. We give general explicit criteria to establish the periodicity of the expansion in both the rational and the quadratic case (for rationals, the qualitative result is due to Laohakosol).

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arXiv2018

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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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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arXiv2015

Rational points on Grassmannians and unlikely intersections in tori

In this paper, we present an alternative proof of a finiteness theorem due to Bombieri, Masser and Zannier concerning intersections of a curve in the multiplicative group of dimension n with algebraic subgroups of dimension n-2. The proof uses a method introduced for the first time by Pila and Zannier to give an alternative proof of Manin-Mumford conjecture and a theorem to count points that satisfy a certain number of linear conditions with rational coefficients. This method has been largely used in many different problems in the context of "unlikely intersections".

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arXiv2014

Upper ramification jumps in abelian extensions of exponent p

In this paper we present a classification of the possible upper ramification jumps for an elementary abelian p-extension of a p-adic field. The fundamental step for the proof of the main result is the computation of the ramification filtration for the maximal elementary abelian p-extension of the base field K. This is a generalization of a previous work of the second author and Dvornicich where the same result is proved under the assumption that K contains a primitive p-th root of unity. Using the class field theory and the explicit relations between the normic group of an extension and its ramification jumps, it is fairly simple to recover necessary and sufficient conditions for the upper ramification jumps of an elementary abelian p-extension of K.

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