# 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
slides or notes for all the talks I have given and my theses (Bachelor,
Master, PhD).

## 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.*
  [arXiv preprint](https://arxiv.org/abs/2111.14363).

* Lombardo, Davide; Tronto, Sebastiano.
  *Some uniform bounds for elliptic curves over Q.*
  [arXiv preprint](https://arxiv.org/abs/2106.09950v1).

* 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).

## Talks

* *Kummer theory for commutative algebraic groups*.
  Seminar at the university of Leiden, September 2022.
  [Slides: [pdf, 1.2Mb](slides-kummer-final.pdf)]

* *My journey in Kummer theory*.
  Leiden-Luxembourg Number Theory PhD Days 2022, July 5-6.
  [Slides: [pdf, 521Kb](slides-journey-kummer.pdf)] 

* *A tour of group theory with our companion cube*.
  University of Luxembourg PhD seminar, May 2022.
  [Slides: [pdf, 1.1Mb](group-cube.pdf)]

* *Kummer theory via (J,T)-extensions*.
  University of Leiden algebra seminar, May 2022;
  similar to *A category of division modules*, see below.

* *Kummer theory for algebraic groups*.
  Nederlands Matematisch Congres, April 2022;
  repeated at the DIAMANT symposium, April 2022.
  [Notes (NMC): [pdf, 215Kb](slides-kummer-kwg.pdf);
   Script (NMC): [txt, 9Kb](script-kummer-kwg.txt);
   Notes (DIAMANT, longer version): [pdf, 223Kb](slides-kummer-diamant.pdf)]

* *Division in modules*.
  University of Leiden algebra seminar, April 2022;
  similar to *A generalization of injective modules* and
  *Division in modules and Kummer theory*, see below.
  [Notes: [pdf, 245Kb](division-leiden.pdf)]

* *Division in modules and Kummer theory*.
  Invited talk at the University of Groningen, February 2022.
  [Slides: [pdf, 249Kb](division-groningen.pdf)]

* *A category of division modules*.
  University of Luxembourg Number Theory seminar, November 2021.
  [Notes: [pdf, 270Kb](notes-division-modules.pdf)]

* *Introduction to étale cohomology*.
  University of Luxembourg seminar series on perverse sheaves, October 2021.
  [Notes: [pdf, 237Kb](notes-etale.pdf)]

* *A generalization of injective modules*.
  University of Luxembourg Number Theory seminar, October 2021.
  [Notes: [pdf, 259Kb](notes-injectivity.pdf)]

* *Integer factorization and elliptic curves*.
  University of Leiden Bachelor seminar (guest talk), April 2021.
  [Slides: [pdf, 393Kb](slides-ecm.pdf)]

* *Group cohomology and elliptic curves*.
  Invited talk at Number Theory Online conference, February 2021.
  [Slides: [pdf, 210Kb](slides-groupcohomec.pdf);
   Video: [link](https://vimeo.com/526814236/1a640285c0?embedded=true&source=video_title&owner=47245911)]

* *Kummer theory for algebraic groups*.
  Invited talk at the University of Bristol (online), October 2020.
  [Slides: [pdf, 239Kb](slides-tronto-bristol.pdf)]

* *Algebraic groups and field extensions*.
  University of Luxembourg PhD seminar (online), April 2020.
  [Slides: [pdf, 280Kb](slides-alggroupsfieldext.pdf)]

* *Field extensions and elliptic curves*.
  University of Luxembourg PhD Day, October 2019.
  [Slides: [pdf, 6.0Mb](slides-fieldextec.pdf)]

* *Kummer theory for elliptic curves*.
  University of Leiden algebra seminar, September 2019;
  repeated with slides at the DIAMANT symposium, November 2019 and
  at Seminari de Teoria de Nombres de Barcelona, February 2020.
  [Notes: [pdf, 181Kb](notes-kummerec.pdf);
   Slides DIAMANT: [pdf, 271Kb](slides-kummec-diamant.pdf);
   Slides Barcelona: [pdf, 290Kb](slides-kummec-barcelona.pdf)].

* *Divisibility of points in algebraic groups*.
  Invited talk at the University of the Basque Country, May 2019.
  [Notes: [pdf, 231Kb](notes-bilbao.pdf)]

* *Kummer theory for number fields*.
  Fifth symposium of the Roman Number Theory association, April 2019;
  repeated at the Algant Alumni symposium in Benasque, May 2019.
  [Slides: [pdf, 291Kb](slides-kummerdegrees.pdf)]

* *The local-global principle*.
  University of Luxembourg interdisciplinary seminar, March 2019.
  [Slides: [pdf, 598Kb](slides-local-global.pdf)]

* *Local-global principle for torsion*.
  University of Luxembourg Number Theory seminar, November 2018.
  [Notes: [pdf, 208Kb](notes-katz.pdf)]