arXiv subjects

Roberto Dvornicich

Publications and source records attributed to Roberto Dvornicich.

6 recordsLinked to original sources

arXiv2024

Classification of rational angles in plane lattices

This paper is concerned with configurations of points in a plane lattice which determine angles that are rational multiples of $π$. We shall study how many such angles may appear in a given lattice and in which positions, allowing the lattice to vary arbitrarily. This classification turns out to be much less simple than could be expected, leading even to parametrizations involving rational points on certain algebraic curves of positive genus.Bulletin of the American Mathematical Society

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arXiv2024

Classification of rational angles in plane lattices II

This paper is a continuation of an earlier one, and completes a classification of the configurations of points in a plane lattice that determine angles that are rational multiples of $π$. We give a complete and explicit description of lattices according to which of these configurations can be found among their points.

math.NT↗

arXiv2022

Local-global questions for divisibility in commutative algebraic groups

This is a survey focusing on the Hasse principle for divisibility of points in commutative algebraic groups and its relation with the Hasse principle for divisibility of elements of the Tate-Shavarevich group in the Weil-Châtelet group. The two local-global subjects arose as a generalization of some classical questions considered respectively by Hasse and Cassels. We describe the deep connection between the two problems and give an overview of the long-established results and the ones achieved during the last twenty years, when the questions were taken up again in a more general setting. In particular, by connecting various results about the two problems, we describe how some recent developments in the first of the two local-global questions imply an answer to Cassel's question, which improves all the results published before about that problem. This answer is best possible over $\mathbb{Q}$. We also describe some links with other similar questions, as for examples the Support Problem and the local-global principle for existence of isogenies of prime degree in elliptic curves.

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arXiv2021

On the division fields of an elliptic curve and an effective bound to the hypotheses of the local-global divisibility

We investigate some aspects of the $m$-division field $K({\mathcal{E}}[m])$, where $\mathcal{E}$ is an elliptic curve defined over a field $K$ with ${\textrm{char}}(K)\neq 2,3$ and $m$ is a positive integer. When $m=p^r$, with $p\geq 5$ a prime and $r$ a positive integer, we prove $K(\mathcal{E}[p^r])=K(x_1,ζ_p,y_2)$, where $\{(x_1, y_1),(x_2,y_2)\}$ is a generating system of ${\mathcal{E}}[p^r]$ and $ζ_p$ is a primitive $p$-th root of the unity. If $\mathcal{E}$ has a $K$-rational point of order $p$, then $K(\mathcal{E}[p^r])=K(ζ_{p^r},\sqrt[m_1]{a})$, with $a\in K(ζ_{p^r})$ and $m_1|p^r$. In addition, when $K$ is a number field, we produce an upper bound to the logarithmic height of the discriminant of the extension $K(\mathcal{E}[m])/K$, for all $m\geq 3$. As a consequence, we give an explicit effective version of the hypotheses of the local-global divisibility problem in elliptic curves over number fields.

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arXiv2017

On Fuchs' Problem about the group of units of a ring

In \cite[Problem 72]{Fuchs60} Fuchs posed the problem of characterizing the groups which are the groups of units of commutative rings. In the following years, some partial answers have been given to this question in particular cases. In a previous paper \cite{DDcharp} we dealt with finite characteristic rings. In this paper we consider Fuchs' question for finite groups and we address this problem in two cases. Firstly, we study the case of torson-free rings and we obtain a complete classification of the finite groups of units which arise in this case. Secondly, we examine the case of characteristic zero rings obtaining, a pretty good description of the possible groups of units equipped with families of examples of both realizable and non-realizable groups. The main tools to deal with this general case are the Pearson and Schneider splitting of a ring \cite{PearsonSchneider70}, our previous results on finite characteristic rings \cite{DDcharp} and our classification of the groups of units of torsion-free rings. As a consequence of our results we completely answer Ditor's question \cite{ditor} on the possible cardinalities of the group of units of a ring.

math.AC↗

arXiv2014

On the integral values of a curious recurrence

We discuss a problem initially thought for the Mathematical Olympiad but which has several interpretations. The recurrence sequences involved in this problem may be generalized to recurrence sequences related to a much larger set of diophantine equations.

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