Research in Number Theory

Chi cerca trova, chi ricerca ritrova (Ennio De Giorgi).

I work in the areas of Number Theory and Algebraic Geometry because I study algebraic groups defined over number fields.
The aim of my research is twofold: firstly, understanding how the properties of algebraic groups and their points can be (algorithmically) detected from their reductions; secondly, making progress in Kummer theory.
From a review: Kummer extensions of number fields, a topic of significant classical interest and enduringly broad appeal in number theory.

Publications

  1. On the order of the reductions of points on abelian varieties and tori, Ph.D. Thesis (defended in 2009), pdf
  2. Prescribing valuations of the order of a point in the reductions of abelian varieties and tori, J. Number Theory, vol. 129 (2009), no.2, 469-476.
  3. Two variants of the support problem for products of abelian varieties and tori, J. Number Theory, vol. 129 (2009), no.8, 1883-1892.
  4. with Peter Jossen, A counterexample to the local-global principle of linear dependence for Abelian varieties , C. R. Acad. Sci. Paris, Ser. I 348 (2010), no.1, 9–10. Presented by J-P. Serre.
  5. On the problem of detecting linear dependence for products of abelian varieties and tori, Acta Arith., vol. 142 (2010), no.2, 119-128.
  6. L'ordine dei punti nelle riduzioni di varietà abeliane e tori, Unione Matematica Italiana, La Matematica nella società e nella cultura, Serie I, vol. 3 (2010), no.1, 128-131.
  7. On the reduction of points on abelian varieties and tori, Int. Math. Res. Notices, vol. 2011 (2011), no.7, 293-308.
  8. The multilinear support problem for products of abelian varieties and tori, Int. J. Number Theory, vol. 8 (2012), no.1, 1-10.
  9. with Chris Hall, On the prime divisors of the number of points on an elliptic curve, C. R. Acad. Sci. Paris, Ser. I 351 (2013) 1-3. Presented by J-P. Serre.
  10. with Jeroen Demeyer, The constant of the support problem for abelian varieties, J. Number Theory, vol. 133 (2013), no.8, 2843-2856.
  11. with Chris Hall, Characterizing abelian varieties by the reductions of the Mordell-Weil group, Pacific J. Math., vol. 265 (2013), no.2, 427-440.
  12. The order of the reductions of an algebraic integer, J. Number Theory, vol. 148 (2015), 121-136.
  13. The prime divisors of the number of points on abelian varieties, J. Théor. Nombres Bordeaux, vol. 27 no. 3 (2015), 805-814.
  14. with Christophe Debry, Reductions of algebraic integers, J. Number Theory, vol. 167 (2016), no.1, 259-283.
  15. Reductions of one-dimensional tori, Int. J. Number Theory, vol. 13 (2017), no. 1, 1473-1489.
  16. with Davide Lombardo, The 1-eigenspace for matrices in GL2(ℤℓ), New York J. Math., vol. 23 (2017), 897-925.
  17. Reductions of algebraic integers II, I.I. Bouw et al. (eds.), Women in Numbers Europe II, Association for Women in Mathematics Series 11 (2018), 10-33.
  18. Reductions of one-dimensional tori II, I.I. Bouw et al. (eds.), Women in Numbers Europe II, Association for Women in Mathematics Series 11 (2018), 35-37.
  19. with Pietro Sgobba, Kummer theory for number fields and the reductions of algebraic numbers, Int. J. Number Theory, vol. 15, No. 8 (2019), 1617--1633.
  20. Multiplicative order and Frobenius symbol for the reductions of number fields, J.S. Balakrishnan et al. (eds.), Research Directions in Number Theory, Association for Women in Mathematics, Series 19 (2019), 161-171.
  21. The problem of detecting linear dependence, Riv. Mat. Univ. Parma, vol. 10 (2019), 99-116.
  22. Reductions of points on elliptic curves, to appear in the Proceedings of the Roman Number Theory Association, vol. 4, no. 1 (2019).
  23. with Pietro Sgobba and Sebastiano Tronto, Kummer theory for number fields, to appear in the Proceedings of the Roman Number Theory Association, vol. 5, no. 1 (2020).
  24. with Pietro Sgobba and Sebastiano Tronto, Addendum to: Reductions of algebraic integers, J. Number Theory, vol. 209 (2020), 391-395.
  25. with Pietro Sgobba, Kummer theory for number fields and the reductions of algebraic numbers II, Uniform Distribution Theory, vol. 15 (2020), no. 1, 75-92.
  26. with Davide Lombardo, Reductions of points on algebraic groups, Institute of Mathematics of Jussieu, DOI: https://doi.org/10.1017/S1474748019000598.
  27. with Peter Bruin, Reductions of points on algebraic groups II, Glas. Math. J., vol. 63, no.2 (2021), 484-502.
  28. with Pietro Sgobba and Sebastiano Tronto, Explicit Kummer Theory for the rational numbers, Int. J. Number Theory, vol. 16 (2020), no. 10, 2213-2231.
  29. with Pietro Sgobba and Sebastiano Tronto, The degree of Kummer extensions of number fields, Int. J. Number Theory, vol. 17, no. 5 (2021), 1091-1110.
  30. The degree of non-Galois Kummer extensions of number fields, Riv. Mat. Univ. Parma, vol. 11 (2020), 301-313.
  31. with Fritz Hörmann, Pietro Sgobba and Sebastiano Tronto, Explicit Kummer theory for quadratic fields, JP J. Algebra, Number Theory Appl., vol. 50 (2021), no. 2, 151-178.
  32. with Pietro Sgobba and Sebastiano Tronto, Kummer Theory for number fields via entanglement groups, waiting for final acceptance in Manuscripta Math..
  33. with Pietro Sgobba and Sebastiano Tronto, Divisibility parameters and Kummer extensions of number fields, waiting for final acceptance in Uniform Distribution Theory.
  34. with Fritz Hörmann, Flavio Perissinotto, Pietro Sgobba, and Sebastiano Tronto, Explicit Kummer generators for cyclotomic extensions, preprint.
  35. with Fritz Hörmann and Flavio Perissinotto, Kummer theory for multiquadratic or quartic cyclic number fields, preprint.

Other projects and Notes

  • Elliptic curves over finite fields with many points (following Waterhouse's 1969 paper), pdf.
  • On the introductory notes on Artin's Conjecture (some details made explicit for students), pdf.
  • Foliazioni semiolomorfe e sottovarietà Levi piatte di una varietà complessa (Master thesis, containing new results), see pdf and the summary pdf.
  • Some variants of the intermediate value theorem for the rationals, pdf.
  • M. Akhim, KU Leuven Master thesis: Elliptic nets and their use in Cryptography (a new algorithm for elliptic nets is developed, the Sage code is included), pdf.
  • C. Debry, KU Leuven Master thesis: Beyond two criteria for supersingularity: coefficients of division polynomials, J. Théorie des Nombres de Bordeaux, vol. 26 (2014), no.3, 595-606, arXiv.
  • by J-P. Serre (Lettre du 3 nov. 2008), A set of prime numbers without a Dirichlet density, pdf.

Coauthors

COLLEAGUES: Peter Bruin, Jeroen Demeyer, Chris Hall, Fritz Hörmann, Peter Jossen, Davide Lombardo.
STUDENTS: Christophe Debry, Flavio Perissinotto, Pietro Sgobba, Sebastiano Tronto.