# Research

After obtaining a bachelor degree in Trento in 2016 I have completed the ALGANT 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 and Peter Bruin. 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; Stellingen: pdf, 123Kb]Master thesis:

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

*Gruppi di Galois e gruppi fondamentali*. University of Trento, 2016. Supervisor: Edoardo Ballico. [pdf, 444Kb]

## Publications and preprints

Tronto, Sebastiano.

*Division in modules and Kummer theory.*arXiv preprint.Lombardo, Davide; Tronto, Sebastiano.

*Some uniform bounds for elliptic curves over Q.*arXiv preprint.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.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]*My journey in Kummer theory*. Leiden-Luxembourg Number Theory PhD Days 2022, July 5-6. [Slides: pdf, 521Kb]*A tour of group theory with our companion cube*. University of Luxembourg PhD seminar, May 2022. [Slides: pdf, 1.1Mb]*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; Script (NMC): txt, 9Kb; Notes (DIAMANT, longer version): pdf, 223Kb]*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 in modules and Kummer theory*. Invited talk at the University of Groningen, February 2022. [Slides: pdf, 249Kb]*A category of division modules*. University of Luxembourg Number Theory seminar, November 2021. [Notes: pdf, 270Kb]*Introduction to étale cohomology*. University of Luxembourg seminar series on perverse sheaves, October 2021. [Notes: pdf, 237Kb]*A generalization of injective modules*. University of Luxembourg Number Theory seminar, October 2021. [Notes: pdf, 259Kb]*Integer factorization and elliptic curves*. University of Leiden Bachelor seminar (guest talk), April 2021. [Slides: pdf, 393Kb]*Group cohomology and elliptic curves*. Invited talk at Number Theory Online conference, February 2021. [Slides: pdf, 210Kb; Video: link]*Kummer theory for algebraic groups*. Invited talk at the University of Bristol (online), October 2020. [Slides: pdf, 239Kb]*Algebraic groups and field extensions*. University of Luxembourg PhD seminar (online), April 2020. [Slides: pdf, 280Kb]*Field extensions and elliptic curves*. University of Luxembourg PhD Day, October 2019. [Slides: pdf, 6.0Mb]*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; Slides DIAMANT: pdf, 271Kb; Slides Barcelona: pdf, 290Kb].*Divisibility of points in algebraic groups*. Invited talk at the University of the Basque Country, May 2019. [Notes: pdf, 231Kb]*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]*The local-global principle*. University of Luxembourg interdisciplinary seminar, March 2019. [Slides: pdf, 598Kb]*Local-global principle for torsion*. University of Luxembourg Number Theory seminar, November 2018. [Notes: pdf, 208Kb]