arXiv subjects

Luca Demangos

Publications and source records attributed to Luca Demangos.

6 recordsLinked to original sources

arXiv2026

Procounting measures and the Bateman--Horn conjecture

Let $D$ be the ring of $S$-integers in a global field and $\da$ its profinite completion. We propose a profinite version of the Bateman--Horn conjecture over $D$ and provide a first comparison with the classical one and its generalizations. Our approach is based on the new notion of procounting measure: a distribution on $\da$ which should be seen as a profinite analogue of the counting function for a subset of $\R$. This allows us to deal with subsets of $\da$ having Haar measure $0$ (corresponding to density zero in $\R$).

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arXiv2020

Densities on Dedekind domains, completions and Haar measure

Let $D$ be the ring of $S$-integers in a global field and $\hat{D}$ its profinite completion. We discuss the relation between density in $D$ and the Haar measure of $\hat{D}$: in particular, we ask when the density of a subset $X$ of $D$ is equal to the Haar measure of its closure in $\hat{D}$. In order to have a precise statement, we give a general definition of density which encompasses the most commonly used ones. Using it we provide a necessary and sufficient condition for the equality between density and measure which subsumes a criterion due to Poonen and Stoll. In another direction, we extend the Davenport-Erd\H{o}s theorem to every $D$ as above and offer a new interpretation of it as a "density=measure" result. Our point of view also provides a simple proof that in any $D$ the set of elements divisible by at most $k$ distinct primes has density 0 for any natural number $k$. Finally, we show that the closure of the set of prime elements of $D$ is the union of the group of units of $\hat{D}$ with a negligible part.

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arXiv2018

Lehmer problem and Drinfeld modules

We propose a lower bound estimate in Dobrowolski's style of the canonical height on a certain family of Drinfeld modules of characteristic 0, including under some hypothesis on their degree and their base field, the complex multiplication case, extending so a previous result of L. Denis on Carlitz modules. Our study is focused on the highest possible level of precision on the parameters involved with rapport to the main values which characterize the Drinfeld module (height, base field degree and rank) and it provides an estimate in function of both separable and inseparable degree of the algebraic points.

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arXiv2017

A few remarks on a Manin-Mumford conjecture in function field arithmetic and generalized Pila-Wilkie estimates

We present here the natural extension of our Pila-Wilkie type estimates on the number of rational points of the trascendent part of a compact analytic subset of $\mathbb{F}_{q}((1/T))^{n}$ to analogous subsets of $K^{n}$, where $K$ is a general local field of any characteristic. That would integrate the analogous estimate provided by F. Loeser, G. Comte and R. Cluckers in characteristic 0.

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arXiv2015

T-modules and Pila-Wilkie estimates

The $T-$modules, introduced by G. Anderson in the '80s, are the natural analogue of abelian varieties in Function Field Arithmetic in positive characteristic. For a special class of them we highlight that a totally similar description of the classic Weierstrass function still holds. In particular, torsion points correspond modulo a finite-rank lattice to rational points of the tangent space. We present in this work an upper bound estimate of the number of rational points of the trascendent part of the analytic set corresponding, into the tangent space, to a nontrivial algebraic subvariety of a $T-$module of this special class. Such an estimate, which takes the same shape of that proved by J. Pila and J. Wilkie, represents our first step of our strategy to prove Manin-Mumford conjecture for such special $T-$modules, based on the ideas developped by U. Zannier and J. Pila.

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arXiv2015

Some examples toward a Manin-Mumford conjecture for T-modules

The aim of this work is to present a possible adaptation of the Manin-Mumford conjecture to the $T-$modules, a mathematical object which has been introduced in the 1980's by G. Anderson as the natural analogue of the abelian varieties in the context of modules over rings which are contained in positive characteristic function fields. We propose then a generalisation of such an adapted conjecture to a modified general version of Mordell-Lang conjecture for $T-$modules which might correct the one proposed for the first time by L. Denis but no longer compatible with the present results.

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