Abelian varieties over ${𝔽}_2$ of prescribed order

Kedlaya, Kiran S.

Publications mathématiques de Besançon. Algèbre et théorie des nombres (2024), pp. 43-58

We prove that for every positive integer $m$, there exist infinitely many simple abelian varieties over ${𝔽}_2$ of order $m$. The method is constructive, building on the work of Madan–Pal in the case $m=1$ to produce an explicit sequence of Weil polynomials giving rise to abelian varieties over ${𝔽}_2$ of order $m$. This sequence itself depends on the choice of a suitable generalized binary representation of $m$; by making careful choices of this representation, we can ensure that the resulting sequence of polynomials have $2$-adic Newton polygons which guarantee the existence of suitable irreducible factors.

Nous démontrons que pour tout entier positif $m$, il existe une infinité de variétés abéliennes simples sur ${𝔽}_2$ d’ordre $m$. La méthode est constructive, se basant sur le travail de Madan–Pal pour le cas $m=1$ pour produire une suite explicite de polynômes de Weil donnant lieu à des variétés abéliennes sur ${𝔽}_2$ d’ordre $m$. La suite elle-même dépend du choix d’une représentation binaire généralisée de $m;$ par des choix soigneux de cette représentation, nous nous assurons que les polynômes qui en résultent ont des polygones de Newton $2$-adiques garantissant l’existence de facteurs irréductibles convenables.

[1] van Bommel, Raymond; Costa, Edgar; Li, Wanlin; Poonen, Bjorn; Smith, Alexander Abelian varieties of prescribed order over finite fields (2021) (https://arxiv.org/abs/2106.13651)

[2] DiPippo, Stephen A.; Howe, Everett W. Real polynomials with all roots on the unit circle and abelian varieties over finite fields, J. Number Theory, Volume 73 (1998) no. 2, pp. 426-450 (corrigendum in [3]) | DOI | MR | Zbl

[3] DiPippo, Stephen A.; Howe, Everett W. Corrigendum: “Real polynomials with all roots on the unit circle and abelian varieties over finite fields”, J. Number Theory, Volume 83 (2000) no. 1, p. 192 | DOI | MR

[4] D’Nelly-Warady, Toren; Kedlaya, Kiran S. Geometric decomposition of abelian varieties of order 1 (2022) (https://arxiv.org/abs/2109.03986v2)

[5] Dupuy, Taylor; Kedlaya, Kiran S.; Roe, David; Vincent, Christelle Isogeny classes of abelian varieties over finite fields in the LMFDB, Arithmetic Geometry, Number Theory, and Computation (Simons Symposia), Springer, 2022, pp. 375-448 | MR

[6] Howe, Everett W.; Kedlaya, Kiran S. Every positive integer is the order of an ordinary abelian variety over ${𝔽}_2$, Res. Number Theory, Volume 7 (2021) no. 4, 59, 6 pages | MR | Zbl

[7] Kadets, Borys Estimates for the number of rational points on simple abelian varieties over finite fields, Math. Z., Volume 297 (2021) no. 1-2, pp. 465-473 | DOI | MR | Zbl

[8] Kedlaya, Kiran S. Search techniques for root-unitary polynomials, Computational Arithmetic Geometry (Contemporary Mathematics), Volume 463, American Mathematical Society, 2008, pp. 71-82 | Zbl | DOI | MR

[9] Madan, Manohar L.; Pal, Sat Abelian varieties and a conjecture of R. M. Robinson, J. Reine Angew. Math., Volume 291 (1977), pp. 78-91 | DOI | MR

[10] Madan, Manohar L.; Queen, Clifford S. Algebraic function fields of class number one, Acta Arith., Volume 20 (1972), pp. 423-432 | DOI | MR | Zbl

[11] Marseglia, Stefano; Springer, Caleb Every finite abelian group is the group of rational points of an ordinary abelian variety over ${𝔽}_2$, ${𝔽}_3$, and ${𝔽}_5$, Proc. Am. Math. Soc., Volume 151 (2023), pp. 501-510 | Zbl | DOI

[12] Mercuri, Pietro; Stirpe, Claudio Classification of algebraic function fields with class number one, J. Number Theory, Volume 154 (2015), pp. 365-374 | DOI | MR | Zbl

[13] Milne, James S. The Riemann hypothesis over finite fields, from Weil to the present day, The Legacy of Bernhard Riemann after One Hundred and Fifty Years. Vol. II (Advanced Lectures in Mathematics), Volume 35, International Press, 2016, pp. 487-565 | MR | Zbl

[14] Reitwiesner, George W. Binary arithmetic, Adv. Comput., Volume 1 (1960), pp. 231-308 | DOI | MR

[15] Robinson, Raphael M. Intervals containing infinitely many sets of conjugate algebraic units, Ann. Math., Volume 80 (1964), pp. 411-428 | DOI | MR | Zbl

[16] Robinson, Raphael M. Conjugate algebraic units in a special interval, Math. Z., Volume 154 (1977), pp. 31-40 | MR | Zbl | DOI

[17] The Sage Developers SageMath, the Sage Mathematics Software System (Version 9.6), 2022 (http://www.sagemath.org/)

[18] Shen, Qibin; Shi, Shuhui Function fields of class number one, J. Number Theory, Volume 154 (2015), pp. 375-379 | Zbl | DOI | MR

[19] Stirpe, Claudio A counterexample to “Algebraic function fields with small class number”, J. Number Theory, Volume 143 (2014), pp. 402-404 | DOI | MR | Zbl

[20] Tate, John Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T. Honda), Séminaire Bourbaki. Vol. 1968/69 (Lecture Notes in Mathematics), Volume 1968, Springer, 1968, pp. 347-363 | MR | Zbl

[21] Waterhouse, William C.; Milne, James S. Abelian varieties over finite fields, 1969 Number Theory Institute (Proceedings of Symposia in Pure Mathematics), Volume 20, American Mathematical Society, 1971, pp. 53-64 | DOI | Zbl