arXiv subjects

Ilaria Del Corso

Publications and source records attributed to Ilaria Del Corso.

7 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 $Γ\subseteq \mathbb{F}^\times$, and a rational prime $p$, we give a general criterion ensuring that $\mathbb{F}(Γ^{p-\mathrm{div}})\setminus Γ^{\mathrm{div}}$ has the Bogomolov property. This problem is motivated by a conjecture of Rémond, 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 Rémond'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.

math.NT↗

arXiv2022

Module braces: relations between the additive and the multiplicative groups

In this paper we define a class of braces, that we call module braces or $R$-braces, which are braces for which the additive group has also a module structure over a ring $R$, and for which the values of the gamma functions are automorphisms of $R$-modules. This class of braces has already been considered in the literature in the case where the ring $R$ is a field: we generalise the definition to any ring $R$, reinterpreting it in terms of the so-called gamma function associated to the brace, and prove that this class of braces enjoys all the natural properties one can require. We exhibit explicit example of R-braces, and we study the splitting of a module braces in relation to the splitting of the ring $R$, generalising thereby Byott's result on the splitting of a brace with nilpotent multiplicative group as a sum of its Sylow subgroups. The core of the paper is in the last two sections, in which, using methods from commutative algebra and number theory, we study the relations between the additive and the multiplicative groups of an $R$-brace showing that if a certain decomposition of the additive group is \emph{small} (in some sense which depends on $R$), then the additive and the multiplicative groups have the same number of element of each order. In some cases, this result considerably broadens the range of applications of the results already known on this issue.

math.GR↗

arXiv2021

On the ranks of the additive and the multiplicative groups of a brace

In \cite[Theorem 2.5]{Bac16} Bachiller proved that if $(G, \cdot, \circ)$ is a brace of order the power of a prime $p$ and the rank of $(G,\cdot)$ is smaller than $p-1$, then the order of any element is the same in the additive and multiplicative group. This means that in this case the isomorphism type of $(G,\circ)$ determines the isomorphism type of $(G,\cdot)$. In this paper we complement Bachiller's result in two directions. In Theorem 2.2 we prove that if $(G, \cdot, \circ)$ is a brace of order the power of a prime $p$, then $(G,\cdot)$ has small rank (i.e. $< p-1$) if and only if $(G,\circ)$ has small rank. We also provide examples of groups of rank $p-1$ in which elements of arbitrarily large order in the additive group become of prime order in the multiplicative group. When the rank is larger, orders may increase.

math.GR↗

arXiv2020

How far is an extension of $p$-adic fields from having a normal integral basis?

Let $L/K$ be a finite Galois extension of $p$-adic fields with group $G$. It is well-known that $\mathcal{O}_L$ contains a free $\mathcal{O}_K[G]$-submodule of finite index. We study the minimal index of such a free submodule, and determine it exactly in several cases, including for any cyclic extension of degree $p$ of $p$-adic fields.

math.NT↗

arXiv2019

Finitely generated abelian groups of units

In 1960 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 this paper we address Fuchs' question for {\it finitely generated abelian} groups and we consider the problem of characterizing those groups which arise in some fixed classes of rings $\mathcal C$, namely the integral domains, the torsion free rings and the reduced rings. To determine the realizable groups we have to establish what finite abelian groups $T$ (up to isomorphism) occur as torsion subgroup of $A^*$ when $A$ varies in $\mathcal C$, and on the other hand, we have to determine what are the possible values of the rank of $A^*$ when $(A^*)_{tors}\cong T$. Most of the paper is devoted to the study of the class of torsion-free rings, which needs a substantially deeper study.

math.RA↗

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

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.

math.NT↗