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# 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 am now working on my PhD
under the supervision of [Antonella Perucca](http://antonellaperucca.net) and
[Peter Bruin](https://www.math.leidenuniv.nl/~pbruin).
## 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
* *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)]
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