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
- On the order of the reductions of points on abelian varieties and tori, Ph.D. Thesis (defended in 2009), pdf
- 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.
- Two variants of the support problem for products of abelian varieties and tori, J. Number Theory, vol. 129 (2009), no.8, 1883-1892.
- 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.
- On the problem of detecting linear dependence for products of abelian varieties and tori, Acta Arith., vol. 142 (2010), no.2, 119-128.
- 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.
- On the reduction of points on abelian varieties and tori, Int. Math. Res. Notices, vol. 2011 (2011), no.7, 293-308.
- The multilinear support problem for products of abelian varieties and tori, Int. J. Number Theory, vol. 8 (2012), no.1, 1-10.
- 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.
- with Jeroen Demeyer, The constant of the support problem for abelian varieties, J. Number Theory, vol. 133 (2013), no.8, 2843-2856.
- with Chris Hall, Characterizing abelian varieties by the reductions of the Mordell-Weil group, Pacific J. Math., vol. 265 (2013), no.2, 427-440.
- The order of the reductions of an algebraic integer, J. Number Theory, vol. 148 (2015), 121-136.
- The prime divisors of the number of points on abelian varieties, J. Théor. Nombres Bordeaux, vol. 27 no. 3 (2015), 805-814.
- with Christophe Debry, Reductions of algebraic integers, J. Number Theory, vol. 167 (2016), no.1, 259-283.
- Reductions of one-dimensional tori, Int. J. Number Theory, vol. 13 (2017), no. 1, 1473-1489.
- with Davide Lombardo, The 1-eigenspace for matrices in GL_{2}(ℤ_{ℓ}),
New York J. Math., vol. 23 (2017), 897-925.
- 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.
- 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.
- 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.
- 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.
- The problem of detecting linear dependence, Riv. Mat. Univ. Parma, vol. 10 (2019), 99-116.
- Reductions of points on elliptic curves, to appear in the Proceedings of the Roman Number Theory Association, vol. 4, no. 1 (2019).
- 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).
- with Pietro Sgobba and Sebastiano Tronto, Addendum to: Reductions of algebraic integers, J. Number Theory, vol. 209 (2020), 391-395.
- 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.
- with Davide Lombardo, Reductions of points on algebraic groups, Institute of Mathematics of Jussieu, DOI: https://doi.org/10.1017/S1474748019000598.
- with Peter Bruin, Reductions of points on algebraic groups II, Glas. Math. J., vol. 63, no.2 (2021), 484-502.
- with Pietro Sgobba and Sebastiano Tronto, Explicit Kummer Theory for the rational numbers, Int. J. Number Theory, vol. 16 (2020), no. 10, 2213-2231.
- 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.
- The degree of non-Galois Kummer extensions of number fields, Riv. Mat. Univ. Parma, vol. 11 (2020), 301-313.
- 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.
- with Pietro Sgobba and Sebastiano Tronto, Kummer Theory for number fields via entanglement groups, waiting for final acceptance in Manuscripta Math..
- with Pietro Sgobba and Sebastiano Tronto, Divisibility parameters and Kummer extensions of number fields, waiting for final acceptance in Uniform Distribution Theory.
- with Fritz Hörmann, Flavio Perissinotto, Pietro Sgobba, and Sebastiano Tronto, Explicit Kummer generators for cyclotomic extensions, preprint.
- 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.