arXiv subjects

Lea Terracini

Publications and source records attributed to Lea Terracini.

At least 19 recordsLinked to original sources

arXiv2026

Small points in radical extensions of number fields

We study small points in radical extensions of algebraic fields. Given an algebraic extension $\mathbb{F}$ of $\mathbb{Q}$, a finitely generated subgroup $\Gamma\subseteq \mathbb{F}^\times$, and a rational prime $p$, we give a general criterion ensuring that $\mathbb{F}(\Gamma^{p-\mathrm{div}})\setminus \Gamma^{\mathrm{div}}$ has the Bogomolov property. This problem is motivated by a conjecture of R\'emond, formulated when $\mathbb{F}$ is a number field, predicting that such radical extensions contain no unexpected small points outside the divisible hull of the group used to generate them. As applications, we obtain new cases of R\'emond's conjecture for radical extensions generated by division points with respect to a finite set of primes, recovering and extending previous results of Amoroso and the third author. Our argument is based on a recent result on small points in $p$-adic Lie extensions.

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arXiv2026

The Bogomolov Property through Galois Representations

The Bogomolov property \B for an algebraic extension of $\QQ$ asserts the existence of a uniform positive lower bound for the absolute logarithmic Weil height outside the group of roots of unity. Originally introduced as a weakening of Northcott's property and closely related to Lehmer's conjecture, it has been established for several natural classes of infinite extensions, including abelian extensions and fields generated by torsion points of elliptic curves defined over the rationals. Given a Galois representation of an absolute Galois group, one can associate with it the algebraic extension fixed by its kernel and ask whether this extension has property \B. This point of view allows one to reinterpret classical results, and to generalize them to other Galois representations, both of geometric and non-geometric origin. This expository paper gives a survey of the techniques and results in this framework. We present some results on modular representations, with a particular focus on the role of local $p$-adic information. We explain how Sen's theorem on totally ramified $p$-adic Lie extensions enters the proof of new criteria for the Bogomolov property, and how these criteria apply to representations with large local image. This contribution is based on joint work with Francesco Amoroso, Andrea Conti, and Pietro Piras.

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arXiv2026

The Bogomolov property for $p$-supercuspidal eigenforms

We prove a lower bound on the Weil height, the so-called Bogomolov property, for the algebraic extensions of $\mathbb Q$ cut out by the adelic Galois representations attached to certain eigenforms whose local component at a prime $p$ is supercuspidal. To this end, we give a method for constructing metric inequalities over $p$-adic Lie extensions of fields over $\mathbb Q$ that are finitely ramified at $p$.

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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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arXiv2025

Bogomolov property for Galois representations with big local image

An algebraic extension of the rational numbers is said to have the $\textit{Bogomolov property}$ (B) if the absolute logarithmic Weil height of its non-torsion elements is uniformly bounded from below. Given a continuous representation $\rho$ of the absolute Galois group $G_{\mathbb{K}}$ of a number field ${\mathbb{K}}$, one says that $\rho$ has (B) if the subfield of $\overline{\mathbb{Q}}$ fixed by $\mathrm{ker}\,\rho$ has (B). We prove that, if $\rho:G_{\mathbb{K}} \to \mathrm{GL}_d({\mathbb{Z}}_p)$ maps an inertia subgroup at a prime above $p$ surjectively onto an open subgroup of $\mathrm{GL}_d({\mathbb{Z}}_p)$, then $\rho$ has (B). More generally, we show that if the image of inertia is open in the image of the decomposition group, the normal closure of the local image is sufficiently large in the global one, and a certain condition on the center of $\rho(G_{\mathbb{K}})$ satisfied, then $\rho$ has (B). In particular, no assumption on the modularity of $\rho$ is needed, contrary to previous work of Habegger and Amoroso--Terracini. We provide several examples both in modular and non-modular cases. Our methods rely on a result of Sen comparing the ramification and Lie filtrations on the $p$-adic Lie group $\rho(G_{\mathbb{K}})$.

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arXiv2024

Bogomolov property and Galois representations

In 2013 P. Habegger proved the Bogomolov property for the field generated over Q by the torsion points of a rational elliptic curve. We explore the possibility of applying the same strategy of proof to the case of field extensions fixed by the kernel of some modular Galois representations.

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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

Large Algebraic Integers

An algebraic integer is said large if all its real or complex embeddings have absolute value larger than $1$. An integral ideal is said \emph{large} if it admits a large generator. We investigate the notion of largeness, relating it to some arithmetic invariants of the field involved, such as the regulator and the covering radius of the lattice of units. We also study its connection with the Weil height and the Bogomolov property. We provide an algorithm for testing largeness and give some applications to the construction of floor functions arising in the theory of continued fractions.

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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

Toric varieties and Gr\"obner bases: the complete Q-factorial case

We present two algorithms determining all the complete and simplicial fans admitting a fixed non-degenerate set of vectors $V$ as generators of their 1-skeleton. The interplay of the two algorithms allows us to discerning if the associated toric varieties admit a projective embedding, in principle for any values of dimension and Picard number. The first algorithm is slower than the second one, but it computes all complete and simplicial fans supported by $V$ and lead us to formulate a topological-combinatoric conjecture about the definition of a fan. On the other hand, we adapt the Sturmfels' arguments on the Gr\"obner fan of toric ideals to our complete case; we give a characterization of the Gr\"obner region and show an explicit correspondence between Gr\"obner cones and chambers of the secondary fan. A homogenization procedure of the toric ideal associated to $V$ allows us to employing GFAN and related software in producing our second algorithm. The latter turns out to be much faster than the former, although it can compute only the projective fans supported by $V$. We provide examples and a list of open problems. In particular we give examples of rationally parametrized families of $\Q$-factorial complete toric varieties behaving in opposite way with respect to the dimensional jump of the nef cone over a special fibre.

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arXiv2019

Simultaneous approximations to p-adic numbers and algebraic dependence via multidimensional continued fractions

Unlike the real case, there are not many studies and general techniques for providing simultaneous approximations in the field of $p$--adic numbers $\mathbb Q_p$. Here, we study the use of multidimensional continued fractions (MCFs) in this context. MCFs were introduced in $\mathbb R$ by Jacobi and Perron as a generalization of continued fractions and they have been recently defined also in $\mathbb Q_p$. We focus on the dimension two and study the quality of the simultaneous approximation to two $p$-adic numbers provided by $p$-adic MCFs, where $p$ is an odd prime. Moreover, given algebraically dependent $p$--adic numbers, we see when infinitely many simultaneous approximations satisfy the same algebraic relation. This also allows to give a condition that ensures the finiteness of the $p$--adic Jacobi--Perron algorithm when it processes some kinds of $\mathbb Q$--linearly dependent inputs.

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arXiv2019

On the finiteness and periodicity of the $p$--adic Jacobi--Perron algorithm

Multidimensional continued fractions (MCFs) were introduced by Jacobi and Perron in order to obtain periodic representations for algebraic irrationals, as it is for continued fractions and quadratic irrationals. Since continued fractions have been also studied in the field of $p$--adic numbers $\mathbb Q_p$, also MCFs have been recently introduced in $\mathbb Q_p$ together to a $p$--adic Jacobi--Perron algorithm. In this paper, we address th study of two main features of this algorithm, i.e., finiteness and periodicity. In particular, regarding the finiteness of the $p$--adic Jacobi--Perron algorithm our results are obtained by exploiting properties of some auxiliary integer sequences. Moreover, it is known that a finite $p$--adic MCF represents $\mathbb Q$--linearly dependent numbers. We see that the viceversa is not always true and we prove that in this case infinite partial quotients of the MCF have $p$--adic valuations equal to $-1$. Finally, we show that a periodic MCF of dimension $m$ converges to algebraic irrationals of degree less or equal than $m+1$ and for the case $m=2$ we are able to give some more detailed results.

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arXiv2018

On p-adic Multidimensional Continued Fractions

Multidimensional continued fractions (MCFs) were introduced by Jacobi and Perron in order to generalize the classical continued fractions. In this paper, we propose an introductive fundamental study about MCFs in the field of the $p$--adic numbers $\mathbb Q_p$. First, we introduce them from a formal point of view, i.e., without considering a specific algorithm that produces the partial quotients of a MCF, and we perform a general study about their convergence in $\mathbb Q_p$. In particular, we derive some conditions about their convergence and we prove that convergent MCFs always strongly converge in $\mathbb Q_p$ contrarily to the real case where strong convergence is not ever guaranteed. Then, we focus on a specific algorithm that, starting from a $m$--tuple of numbers in $\mathbb Q_p$, produces the partial quotients of the corresponding MCF. We see that this algorithm is derived from a generalized $p$--adic Euclidean algorithm and we prove that it always terminates in a finite number of steps when it processes rational numbers.

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arXiv2018

Embedding the Picard group inside the class group: the case of $\Q$-factorial complete toric varieties

Let $X$ be a $\Q$-factorial complete toric variety over an algebraic closed field of characteristic $0$. There is a canonical injection of the Picard group ${\rm Pic}(X)$ in the group ${\rm Cl}(X)$ of classes of Weil divisors. These two groups are finitely generated abelian groups; whilst the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of ${\rm Pic}(X)$ in ${\rm Cl}(X)$ is contained in a free part of the latter group.

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arXiv2017

A $\mathbb{Q}$--factorial complete toric variety with Picard number 2 is projective

This paper is devoted to settle two still open problems, connected with the existence of ample and nef divisors on a Q-factorial complete toric variety. The first problem is about the existence of ample divisors when the Picard number is 2: we give a positive answer to this question, by studying the secondary fan by means of Z-linear Gale duality. The second problem is about the minimum value of the Picard number allowing the vanishing of the Nef cone: we present a 3-dimensional example showing that this value cannot be greater then 3, which, under the previous result, is also the minimum value guaranteeing the existence of non-projective examples.

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