Stefano Marseglia

Papers

Preprints

15. Abelian varieties over finite fields with commutative endomorphism algebra: theory and algorithms
    with Jonas Bergström and Valentijn Karemaker
    arXiv --- Magma code --- code (examples)
16. Abelian surfaces over finite fields containing no curves of genus 3 or less
    with Elena Berardini and Alejandro Giangreco Maidana
    arXiv
17. Local isomorphism classes of fractional ideals of orders in étale algebras
    arXiv --- Magma code

Journals

12. Abelian varieties over finite fields and their groups of rational points
    with Caleb Springer
    to appear in Algebra & Number Theory
    arXiv
13. 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
14. Modules over orders, conjugacy classes of integral matrices, and abelian varieties over finite fields
    to appear in Sixteenth Algorithmic Number Theory Symposium ( ANTS XVI )
    arXiv --- Magma code for examples
15. Cohen-Macaulay type of orders, generators and ideal classes
    to appear in Journal of Algebra
    journal (open access) --- arXiv --- Magma code
8. 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
9. 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 --- code (examples)
10. 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
11. 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 --- code (examples)
12. Computing abelian varieties over finite fields isogenous to a power,
    Res. number theory 5 (2019), no. 4, paper no. 35
    journal (open access) --- arXiv --- code (examples)
13. Computing square-free polarized abelian varieties over finite fields,
    Mathematics of Computation 90 (2021), no. 328, 953–971.
    journal --- arXiv --- code (examples)
14. Computing the ideal class monoid of an order,
    J. Lond. Math. Soc. 101 (2020), no. 3, 984-1007
    journal --- arXiv
15. 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:

• 2.13.e_o
• 3.3.ab_ac_f
• 4.3.d_l_w_by
• 5.2.ab_c_ab_e_af

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 papers.
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 and 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 requires AlgEt.
- 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.
- It is an expansion of the package AbVarFq.
- The package will be updated to use the functionalities developed in AlgEt for increased speed.
- This packages is based on theory developed in Paper [7].

IsomClAbVarFqCommEndAlg - Magma - link
- It provides functionalities to compute (unpolarized) Fq-isomorphism classes of abelian varieties over Fq belonging to an isogeny class with commutative Fq-endomorphism ring, for any finite field Fq. In particular, the code works for any p-rank. - It is an expansion of the package AbVarFq.
- This packages is based on theory developed in Paper [15].

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.