# Research

After obtaining a bachelor degree in Trento in 2016 I have completed the
[ALGANT](https://algant.eu) Master Program (Milano and Leiden), with
special focus on number theory and algebraic geometry. I then moved on to
obtain a joint PhD at the universities of Luxembourg and Leiden, under
the supervision of [Antonella Perucca](http://antonellaperucca.net) and
[Peter Bruin](https://www.math.leidenuniv.nl/~pbruin).

I have collected in this page the list of papers I puplished during my PhD,
as well as theses (Bachelor, Master, PhD). See [my talks page](../talks)
for a list of slides and notes of my talks.

## Theses

* Phd thesis: *Kummer theory for commutative algebraic groups*.
  University of Luxembourg and University of Leiden, 2022.
  Supervisors: Antonella Perucca, Peter J. Bruin.
  [Thesis: [pdf, 1.2Mb](phd-thesis.pdf);
  Stellingen: [pdf, 123Kb](stellingen.pdf)]

* Master thesis: *The Brauer-Manin obstruction to strong approximation*.
  University of Milan and University of Leiden, 2018.
  Supervisor: Martin J. Bright.
  [[pdf, 800Kb](master-thesis.pdf)]

* Bachelor thesis: *Gruppi di Galois e gruppi fondamentali*.
  University of Trento, 2016.
  Supervisor: Edoardo Ballico.
  [[pdf, 444Kb](bachelor-thesis.pdf)]

## Publications and preprints

* Tronto, Sebastiano.
  *Division in modules and Kummer theory.*
  [Journal de Thórie des Nombres de Bordeaux (2025) - open access](https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1326)

* Lombardo, Davide; Tronto, Sebastiano.
  *Some uniform bounds for elliptic curves over Q.*
  [Pacific Journal of Mathematics (2022)](https://msp.org/pjm/2022/320-1/p06.xhtml).

* Lombardo, Davide; Tronto, Sebastiano.
  *Effective Kummer theory for elliptic curves.*
  International Mathematics Research Notices (2021).

* Hörmann, Fritz; Perucca, Antonella; Sgobba, Pietro; Tronto, Sebastiano.
  *Explicit Kummer theory for quadratic fields.*
  JP Journal of Algebra, Number Theory and Applications (2021).

* Perucca, Antonella; Sgobba, Pietro; Tronto, Sebastiano.
  *The degree of Kummer extensions of number fields.*
  International Journal of Number Theory (2021).

* Perucca, Antonella; Sgobba, Pietro; Tronto, Sebastiano.
  *Kummer Theory for number fields via entanglement groups.*
  Manuscripta Mathematica (2021).

* Tronto, Sebastiano.
  *Radical entanglement for elliptic curves.*
  [arXiv preprint](https://arxiv.org/abs/2009.08298).

* Perucca, Antonella; Sgobba, Pietro; Tronto, Sebastiano.
  *Kummer theory for number fields.*
  Proceedings of the Roman Number Theory Association (2020).

* Perucca, Antonella; Sgobba, Pietro; Tronto, Sebastiano.
  *Addendum to: Reductions of Algebraic integers.*
  Journal of Number Theory (2020).

* Perucca, Antonella; Sgobba, Pietro; Tronto, Sebastiano.
  *Explicit Kummer theory for the rational numbers.*
  International Journal of Number Theory (2020).