Stefano Marseglia



Papers

Preprints
  1. Local isomorphism classes of fractional ideals of orders in étale algebras
    arXiv --- Magma code
  2. Modules over orders, conjugacy classes of integral matrices, and abelian varieties over finite fields
    arXiv --- Magma code for examples
Journals
  1. Abelian varieties over finite fields and their groups of rational points
    with Caleb Springer
    to appear in Algebra & Number Theory

    arXiv
  2. Ideal classes of orders in quaternion algebras
    with Harry Smit and an Appendix by John Voight
    J. Pure Appl. Algebra 228 (2024), no. 7, 107649.

    journal (open access) --- arXiv --- Magma code
  3. Cohen-Macaulay type of orders, generators and ideal classes
    to appear in Journal of Algebra arXiv --- Magma code
  4. Every finite abelian group is the group of rational points of an ordinary abelian variety over F2,F3 and F5
    with Caleb Springer
    Proc. Amer. Math. Soc. 151 (2023), no. 2, 501-510.
    journal --- arXiv
  5. Polarizations of abelian varieties over finite fields via canonical liftings,
    with Jonas Bergström and Valentijn Karemaker
    Int. Math. Res. Not. IMRN (2023), no.4, 3194-3248
    journal (open access) --- arXiv --- Magma code --- examples
  6. Products and Polarizations of Super-Isolated Abelian Varieties,
    with Travis Scholl
    Mathematische Zeitschrift, 300 (2022), no. 1, 445–462.
    journal (open access) --- arXiv --- Sage code
  7. Computing base extensions of ordinary abelian varieties over a finite field,
    Int. J. Number Theory, Vol. 18, No. 09, pp. 1957-1974 (2022)
    journal --- arXiv --- Magma code (examples)
  8. Computing abelian varieties over finite fields isogenous to a power,
    Res. number theory 5 (2019), no. 4, paper no. 35
    journal (open access) --- arXiv --- Magma code (examples) file 1, file 2
  9. Computing square-free polarized abelian varieties over finite fields,
    Mathematics of Computation 90 (2021), no. 328, 953–971.
    journal --- arXiv --- code (examples)
  10. Computing the ideal class monoid of an order,
    J. Lond. Math. Soc. 101 (2020), no. 3, 984-1007
    journal --- arXiv
  11. Super-multiplicativity of ideal norms in number fields,
    Acta Arith. 193 (2020), no. 1, 75–93.
    journal --- arXiv
PhD thesis
  • Computing abelian varieties over finite fields,
    PhD thesis, defended on June 8, 2018, permanent link

Data

Data about endomorphism rings and polarizations of abelian varieties over finite fields for isogeny classes of abelian varieties is now on the LMFDB xyz.
Here are some examples:

Computer Packages

Instructions on how to use each package are included in the attached README files.
We invite also to look at the code to produce the examples for the corresponding.
Don't hesitate to contact me if you have questions and/or comments!
EllipticCurves - Macaulay2 - link
- It provides basic functionalities for computations with elliptic curves.
- Written in collaboration with Alessandro Oneto.
- The development is concluded.
AlgEt - Magma - link
- It provides functionalities to compute with orders in étale algebras over Q.
- In particular, one can compute isomorphism classes of modules and the monoid of the ideal classes of the fractional ideals of the order.
- A version of this package has been included in Magma 2.28-1 (without documentation).
- The version available through the link is faster and it will be maintained nd expanded.
- This packages is based on theory developed in Papers [3], [9], [10] and [13].
IdlClQuat - Magma - link
- It provides functionalities to compute the isomorphism classes of ideals of an order in a quaternion algebra.
- This is a collaboration with Harry Smit.
- Currently, there is no plan to expand it.
- This packages is based on theory developed in Paper [11].
AbVarFq - Magma - link
- It provides functionalities to compute isomorphism classes of abelian varieties (in the ordinary case and over a prime field) and polarizations (in the ordinary case).
- The package will be updated to use the functionalities developed in AlgEt for increase speed.
- This packages is based on theory developed in Papers [3], [4] and [5].
PolsAbVarFpCanLift - Magma - link
- It provides functionalities to compute polarizations for squarefree isogeny classes of abelian varieties over a prime field even when they are not ordinary.
- The package will be updated to use the functionalities developed in AlgEt for increase speed.
- This packages is based on theory developed in Paper [7].

Slides/Videos of some of my talks

Cohen-Macaulay type of endomorphism rings of abelian varieties over finite fields - Slides, - AGC²T 2023 , CIRM, Luminy, France. - June 6, 2023.
Polarizations of abelian varieties over finite fields via canonical liftings - Slides, - Video - Utrecht Geometry Centrum, Utrecht University, Netherlands. - Mar 29, 2022.
Computing Isomorphism Classes of Abelian Varieties over Finite Fields - Slides, - Video - VaNTAGe 11th series "Curves and abelian varieties over finite fields" (online). - Feb 1, 2022.
Computing Isomorphism Classes of Abelian Varieties over Finite Fields - Slides - Semi-Plenary Talk: Curves over finite fields - past, present and future (online) - May 24, 2021.
Products and Polarizations of Super-Isolated Abelian Varieties - Slides - (Q)NTAG (online), Simon Fraser University, Burnaby, Canada - December 03, 2020.
Abelian varieties over finite fields isogenous to a power - Slides - Institut de recherche mathématique de Rennes IRMAR, , Rennes, France. - November 16, 2018.