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Jorge Mello

Publications and source records attributed to Jorge Mello.

At least 19 recordsLinked to original sources
arXiv2026

On the classification of small cyclotomic integers

We give a general classification theorem for cyclotomic algebraic integers with all complex absolute values bounded by a fixed constant $c$, modeled on the theorem of Cassels which treats the case $c = \sqrt{5}$ up to finitely many exceptions. As a corollary, we establish that the range of the function taking a cyclotomic integer to its maximum complex absolute value is a well-ordered (but not closed) subset of the real numbers. We also formulate analogous statements for algebraic numbers in the maximal cyclotomic extension of a fixed number field. The proofs combine a result of Loxton, which bounds the number of roots of unity in the shortest additive representation of a cyclotomic integer in terms of the maximum complex absolute value, with an equidistribution theorem of Bilu et al. for Galois orbits of torsion points on algebraic tori.

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arXiv2026

On higher dimensional integrality and multiplicative dependence in semigroup algebraic dynamics

We study multiplicative dependence of points in semigroup orbits in higher dimensions. More specifically, we show that the non-density of integral points in semigroup orbits implies sparsity of multiplicative dependence in orbits. This can be viewed as a semigroup dynamical and a higher dimensional version of recent results by B\'{e}rczes, Ostafe, Shparlinski and Silverman, which in turn can be viewed as a generalization of theorems of Northcott and Siegel. We also confirm that the non-density hypothesis of integral points in orbits is implied by Vojta's conjecture.

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arXiv2026

Hierarchical Locally Recoverable Codes on surfaces

We construct locally recoverable codes with hierarchy from surfaces in $\mathbb{A}^3$ admitting a fibration by curves of Artin-Schreier or Kummer type. We derive the parameters of our codes by leveraging the geometry and arithmetic of the fibration, which is obtained by projection onto one of the coordinates. As a byproduct, we obtain estimates for (and in one case an explicit count of) the number of rational points in certain families of surfaces.

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arXiv2025

The exceptional set in Cassel's theorem on small cyclotomic integers

In a 1965 paper, R. Robinson made five conjectures about the classification of cyclotomic algebraic integers for which the maximum absolute value in any complex embedding (the house) is small, modulo the equivalence relation generated by Galois conjugation and multiplication by roots of unity. In response to one of these conjectures, Cassels showed in 1969 that when the house is at most $\sqrt{5}$, one obtains three parametric families plus an effectively computable finite set of equivalence classes of exceptions. Building on the work of Jones, Calegari-Morrison-Snyder, and Robinson-Wurtz, we determine this exceptional set. By specializing to the case where the house is strictly less than 2, we resolve the final outstanding conjecture from Robinson's 1965 paper.

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arXiv2025

Rational Points and Zeta Functions of Humbert Surfaces with Square Discriminant

This paper examines the arithmetic of the loci \(\cL_n\), parameterizing genus 2 curves with \((n, n)\)-split Jacobians over finite fields \(\F_q\). We compute rational points \(|\cL_n(\F_q)|\) over \(\F_3\), \(\F_9\), \(\F_{27}\), \(\F_{81}\), and \(\F_5\), \(\F_{25}\), \(\F_{125}\), derive zeta functions \(Z(\cL_n, t)\) for \(n = 2, 3\). Utilizing these findings, we explore isogeny-based cryptography, introducing an efficient detection method for split Jacobians via explicit equations, enhanced by endomorphism ring analysis and machine learning optimizations. This advances curve selection, security analysis, and protocol design in post-quantum genus 2 systems, addressing efficiency and vulnerabilities across characteristics.

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arXiv2023

Explicit bounds for the solutions of superelliptic equations over number fields

Let $f$ be a polynomial with coefficients in the ring $O_S$ of $S$-integers of a number field $K$, $b$ a non-zero $S$-integer, and $m$ an integer $\ge 2$. We consider the equation $( \star )$: $f(x) = b y^m$ in $x,y \in O_S$. Under the well-known LeVeque condition, we give fully explicit upper bounds in terms of $K, S, f, m$ and the $S$-norm of $b$ for the heights of the solutions $x$ of the equation $( \star)$. Further, we give an explicit bound $C$ in terms of $K, S, f$ and the $S$-norm of $b$ such that if $m > C$ the equation $(\star)$ has only solutions with $y = 0$ or a root of unity. Our results are more detailed versions of work of Trelina, Brindza, Shorey and Tijdeman, Voutier and Bugeaud, and extend earlier results of B\'erczes, Evertse, and Gy\H{o}ry to polynomials with multiple roots. In contrast with the previous results, our bounds depend on the $S$-norm of $b$ instead of its height.

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arXiv2022

An asymptotic for sums of Lyapunov exponents in families

Let f_t be a meromorphic family of endomorphisms of P^N_C of degree at least 2, and let L(f_t) be the sum of Lyapunov exponents associated to f_t. Favre showed that L(f_t)=L(f)\log|t^{-1}|+o(\log|t^{-1}|) as t -> 0, where L(f) is the sum of Lyapunov exponents on the generic fibre, interpreted as an endomorphism of some projective Berkovich space. Under some additional constraints on the family, we provide an explicit error term.

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arXiv2021

Multiplicative dependence of rational values modulo approximate finitely generated groups

In this paper, we establish some finiteness results about the multiplicative dependence of rational values modulo sets which are `close' (with respect to the Weil height) to division groups of finitely generated multiplicative groups of a number field $K$. For example, we show that under some conditions on rational functions $f_1, \ldots, f_n\in K(X)$, there are only finitely many elements $\alpha \in K$ such that $f_1(\alpha),\ldots,f_n(\alpha)$ are multiplicatively dependent modulo such sets.

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arXiv2020

On effective $\epsilon$-integrality in orbits of rational maps over function fields and multiplicative dependence

We give effective bounds for the set quasi-integral points in orbits of non-isotrivial rational maps over function fields under some conditions, generalizing previous work of Hsia and Silverman (2011) for orbits over function fields of characteristic zero. We then use this to prove finiteness results for algebraic functions whose orbit under a rational function has multiplicative dependent elements modulo rings of $S$-integers, generalizing recent results over number fields.

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arXiv2020

On elements of large order of elliptic curves and multiplicative dependent images of rational functions over finite fields

Let $E_1$ and $E_2$ be elliptic curves in Legendre form with integer parameters. We show there exists a constant $C$ such that for almost all primes, for all but at most $C$ pairs of points on the reduction of $E_1 \times E_2$ modulo $p$ having equal $x$ coordinate, at least one among $P_1$ and $P_2$ has a large group order. We also show similar abundance over finite fields of elements whose images under the reduction modulo $p$ of a finite set of rational functions have large multiplicative orders

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arXiv2020

On abelian points of varieties intersecting subgroups in a torus

We show, under some natural conditions, that the set of abelian points on the non-anomalous subset of a closed irreducible subvariety $X$ intersected with the union of connected algebraic subgroups of codimension at least $\dim X$ in a torus is finite, generalising results of Ostafe, Sha, Shparlinski and Zannier (2017). We also generalise their structure theorem for such sets when the algebraic subgroups are not necessarily connected, and obtain a related result in the context of curves and arithmetic dynamics.

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arXiv2020

An effective local-global principle for algebraic varieties and the sum product problem in finite fields

We use recent results about linking the number of zeros on algebraic varieties over $\mathbb{C}$, defined by polynomials with integer coefficients, and on their reductions modulo sufficiently large primes to study congruences with products and reciprocals of linear forms. This allows us to make some progress towards a question of B. Murphy, G. Petridis, O. Roche-Newton, M. Rudnev and I. D. Shkredov (2019) on an extreme case of the Erd\H{o}s-Szemer\'{e}di conjecture in finite fields.

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arXiv2019

On the properties of Northcott and Narkiewicz for elliptic curves

In this paper, for an elliptic curve $E$ defined over the algebraic numbers and for any subfield $F$ of algebraic numbers, we say that $E$ has the Northcott property over $F$ if there are at most finitely many $F$-rational points on $E$ of uniformly bounded height, and we say that $E$ has the property (P) over $F$ if for any infinite subset $S$ of $F$-rational points on $E$, $f(S) = S$ for an $F$-endomorphism $f$ of $E$ implies that $f$ is an automorphism. We establish some criteria for both properties and provide typical examples. We also show that the Northcott property implies the property (P).

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