Research in Number Theory

Provando e riprovando.

Coauthors

Bryan Advocaat, Félix Baril Boudreau, Alexandre Benoist, Peter Bruin, Chi Wa (Clifford) Chan, Christophe Debry, Jeroen Demeyer, Francesc Fité, Chris Hall, Fritz Hörmann, Olli Järviniemi, Tom Kasel, Peter Jossen, Davide Lombardo, Pieter Moree, Antigona Pajaziti, Flavio Perissinotto, Tim Seuré, Igor Shparlinski, Pietro Sgobba, Mia Tholl, Sebastiano Tronto, Vincent Wolff.

Publications

Reductions of algebraic groups and torsion/Kummer extensions

• 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.
• On the reduction of points on abelian varieties and tori, Int. Math. Res. Notices, vol. 2011 (2011), no.7, 293-308.
• 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 Chris Hall, Characterizing abelian varieties by the reductions of the Mordell-Weil group, Pacific J. Math., vol. 265 (2013), no.2, 427-440.
• The prime divisors of the number of points on abelian varieties, J. Théor. Nombres Bordeaux, vol. 27 no. 3 (2015), 805-814.
• with Davide Lombardo, The 1-eigenspace for matrices in GL₂(ℤ_ℓ), New York J. Math., vol. 23 (2017), 897-925.
• with Davide Lombardo, Reductions of points on algebraic groups, J. Inst. Math. Jussieu vol. 20 (2021), no. 5, 1637-1669.
• with Peter Bruin, Reductions of points on algebraic groups II, Glas. Math. J., vol. 63, no.2 (2021), 484-502.
• with Francesc Fité, A local-global principle for polyquadratic twists of abelian surfaces, to appear in Indiana Univ. Math. J..
• with Antigona Pajaziti, One formal result for torsion and arboreal representations of commutative algebraic groups, preprint.

Detecting linear dependence and the support problem

• On the problem of detecting linear dependence for products of abelian varieties and tori, Acta Arith., vol. 142 (2010), no.2, 119-128.
• 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.
• The problem of detecting linear dependence, Riv. Mat. Univ. Parma, vol. 10 (2019), 99-116.
• Two variants of the support problem for products of abelian varieties and tori, J. Number Theory, vol. 129 (2009), no.8, 1883-1892.
• The multilinear support problem for products of abelian varieties and tori, Int. J. Number Theory, vol. 8 (2012), no.1, 1-10.
• with Jeroen Demeyer, The constant of the support problem for abelian varieties, J. Number Theory, vol. 133 (2013), no.8, 2843-2856.

Kummer theory

• The order of the reductions of an algebraic integer, J. Number Theory, vol. 148 (2015), 121-136.
• with Christophe Debry, Reductions of algebraic integers, J. Number Theory, vol. 167 (2016), no.1, 259-283
  (errata: in Table 3, left side the density for the fifth and seventh entries should both read 121/960)
• Reductions of one-dimensional tori, Int. J. Number Theory, vol. 13 (2017), no. 1, 1473-1489.
• Reductions of algebraic integers II, I.I. Bouw et al. (eds.), Women in Numbers Europe II, Assoc. Women Math. Ser. vol. 11 (2018), 10-33.
• Reductions of one-dimensional tori II, I.I. Bouw et al. (eds.), Women in Numbers Europe II, Assoc. Women Math. Ser. vol. 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, Assoc. Women Math. Ser. vol. 19 (2019), 161-171.
• 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, Unif. Distrib. Theory, vol. 15 (2020), no. 1, 75-92.
• 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, Manuscripta Math., vol. 169 (2022), no. 1-2, 251-270.
• with Pietro Sgobba and Sebastiano Tronto, Divisibility parameters and Kummer extensions of number fields, Unif. Distrib. Theory, vol. 16 (2021), no. 2, 71-88.
• with Fritz Hörmann, Pietro Sgobba, and Sebastiano Tronto, Explicit Kummer generators for cyclotomic extensions, JP J. Algebra, Number Theory Appl., vol. 53 (2022), no. 1, 69-84.
• with Flavio Perissinotto, Kummer theory for multiquadratic or quartic cyclic number fields, Unif. Distrib. Theory, vol. 17 (2022), no. 2, 165-194.
• with Flavio Perissinotto, Kummer theory for products of one-dimensional tori, Publ. Math. Besançon Algèbre Théorie Nr., vol. 2023 (2023), 109-119.
• with Bryan Advocaat, Chi Wa (Clifford) Chan, Antigona Pajaziti, and Flavio Perissinotto, Galois groups of Kummer extensions of number fields, to appear in Publ. Math. Besançon Algèbre Théorie Nr..
• with Flavio Perissinotto, Kummer theory for p-adic fields, preprint.
• with Félix Baril Boudreau, Kummer theory for function fields, preprint.

Artin's conjecture on primitive roots

• with Olli Järviniemi, Unified treatment of Artin-type problems, to appear in Res. Number Theory, vol. 9 (2023), no.10, 20 pages.
• with Pieter Moree and Pietro Sgobba, The distribution of the multiplicative index of algebraic numbers over residue classes, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, published online in 2024.
• with Olli Järviniemi and Pietro Sgobba, Unified treatment of Artin-type problems II, preprint.
• with Pietro Sgobba, Comparing densities for powers in Artin's conjecture on primitive roots, preprint.
• with Igor Shparlinski, Uniform bounds for the density in Artin's conjecture on primitive roots, preprint.
• with Mia Tholl, Computing primitive roots according to Artin's conjecture, to appear in JP J. Algebra, Number Theory Appl..
• with Tom Kasel, Primitive roots, Fermat primes and Sophie Germain primes, preprint.

Miscellanea

• with Tim Seuré and Vincent Wolff, Divisibility Graphs and Modular Multiplication Tables, American Mathematical Monthly, published online in 2024.
• with Alexandre Benoist, The ternary cyclotomic polynomials Φ_3pq, preprint.
• with Flavio Perissinotto, Kummer extensions of finite fields, short note.
• Elliptic curves over finite fields with many points (notes following Waterhouse's 1969 paper), pdf.
• On the introductory notes on Artin's Conjecture (some details made explicit for students), pdf.
• L'ordine dei punti nelle riduzioni di varietà abeliane e tori, Mat. Cult. Soc. Riv. Unione Mat. Ital. (I), vol. 3 (2010), no.1, 128-131.
• M. Akhim, KU Leuven Master thesis: Elliptic nets and their use in Cryptography (a new algorithm for elliptic nets is developed), pdf.
• by J-P. Serre (Lettre du 3 nov. 2008), A set of prime numbers without a Dirichlet density, pdf.