arXiv subjects

Marzio Mula

Publications and source records attributed to Marzio Mula.

5 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

The Hessian of elliptic curves as a Latt\`es map

We prove that the Hessian transformation of elliptic curves, both as an action on $j$-invariants and on the Hesse pencil, is a rigid Latt\`es map fitting into a reduced diagram, hence it lifts to a degree-$3$ endomorphism $\psi$ of a prescribed elliptic curve $E$. This result provides an effective tool to investigate the dynamics of the Hessian transformation, whose symmetries are inherited from those of $\psi$, which we characterize. In particular, over arbitrary fields of characteristic different from $2$ and $3$, the functional graphs of the Hessian and, more generally, of Latt\`es maps fitting into analogous reduced diagrams, are completely determined by the action of $\psi$ on the twists of $E$. When the underlying field is finite, we specialize these results to obtain a complete classification of Hessian functional graphs and derive an efficient method for computing iterated Hessians.

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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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