Geodesic
The characteristic polynomials of geometrically split ordinary abelian varieties of dimension $3$
Prikladnaya Diskretnaya Matematika. Supplement, no. 17 (2024), pp. 12-16.

Voir la notice de l'article provenant de la source Math-Net.Ru

In the paper, we explicitly describe all possible characteristic polynomials of the Frobenius endomorphism for ordinary geometrically decomposable Abelian varieties of dimension $3$ over a finite field. These polynomials encode many arithmetic properties of abelian varieties including number of points. More precisely, if $\chi_{A,{q}}(T)$ is the characteristic polynomial of the Frobenius endomorphism on $A$ over $\mathbb{F}_q$, then the number of points on $A$ is equal to $\chi_{A,{q}}(1)$.
Keywords: Abelian threefold, characteristic polynomial, point-counting, finite field. 
@article{PDMA_2024_17_a2,
     author = {S. A. Novoselov},
     title = {The characteristic polynomials of geometrically split ordinary abelian varieties of dimension $3$},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {12--16},
     publisher = {mathdoc},
     number = {17},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2024_17_a2/}
}
TY  - JOUR
AU  - S. A. Novoselov
TI  - The characteristic polynomials of geometrically split ordinary abelian varieties of dimension $3$
JO  - Prikladnaya Diskretnaya Matematika. Supplement
PY  - 2024
SP  - 12
EP  - 16
IS  - 17
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PDMA_2024_17_a2/
LA  - ru
ID  - PDMA_2024_17_a2
ER  - 
%0 Journal Article
%A S. A. Novoselov
%T The characteristic polynomials of geometrically split ordinary abelian varieties of dimension $3$
%J Prikladnaya Diskretnaya Matematika. Supplement
%D 2024
%P 12-16
%N 17
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PDMA_2024_17_a2/
%G ru
%F PDMA_2024_17_a2
S. A. Novoselov. The characteristic polynomials of geometrically split ordinary abelian varieties of dimension $3$. Prikladnaya Diskretnaya Matematika. Supplement, no. 17 (2024), pp. 12-16. http://geodesic.mathdoc.fr/item/PDMA_2024_17_a2/

[1] Schoof R., “Counting points on elliptic curves over finite fields”, J. de théorie des nombres de Bordeaux, 7:1 (1995), 219–254 | MR | Zbl

[2] Gaudry P., Schost É., “Genus 2 point counting over prime fields”, J. Symbolic Comput., 47:4 (2012), 368–400 | DOI | MR | Zbl

[3] Abelard S., Gaudry P., Spaenlehauer P. J., “Improved complexity bounds for counting points on hyperelliptic curves”, Foundations Comput. Math., 19 (2019), 591–621 | DOI | MR | Zbl

[4] Dobson S., Galbraith S. D., Smith B., “Trustless groups of unknown order with hyperelliptic curves”, Math. Cryptology, 1:2 (2022), 25–39

[5] Howe E. W., Zhu H. J., “On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field”, J. Number Theory, 92:1 (2002), 139–163 | DOI | MR | Zbl

[6] Freeman D. M., Satoh T., “Constructing pairing-friendly hyperelliptic curves using Weil restriction”, J. Number Theory, 131:5 (2011), 959–983 | DOI | MR | Zbl

[7] Chou K. M. J., Kani E., “Simple geometrically split abelian surfaces over finite fields”, J. Ramanujan Math. Soc., 29:1 (2014), 31–62 | MR | Zbl

[8] Novoselov S. A., Boltnev Yu. F., “On the number of points on the curve $y^2=x^7+a x^4 + b x$ over a finite field”, Diskretn. Anal. Issled. Oper., 29:2 (2022), 62–79 (in Russian) | MR | Zbl

[9] Chai C. L., Conrad B., Oort F., Complex Multiplication and Lifting Problems, AMS, Providence, 2013 | MR

[10] Cohen H., Frey G., Avanzi R., et al., Handbook of Elliptic and Hyperelliptic Curve Cryptography, CRC Press, 2005

[11] Arpin S., Camacho-Navarro C., Lauter K., et al., “Adventures in supersingularland”, Experimental Math., 32:2 (2023), 241–268 | DOI | MR | Zbl

[12] Abelian variety isogeny class 3.23.al_dp_asj over $\mathbb{F}_{23}$, , 2024 https://www.lmfdb.org/Variety/Abelian/Fq/3/23/al_dp_asj

[13] Abelian variety isogeny class 3.7.a_a_bl over $\mathbb{F}_{7}$, , 2024 https://www.lmfdb.org/Variety/Abelian/Fq/3/7/a_a_bl