Variations in the distribution of principally polarized abelian varieties among isogeny classes

Howe, Everett W.

Annales Henri Lebesgue, Tome 5 (2022), pp. 677-702.

Nous montrons que pour une grande classe d’anneaux $R$, le nombre de variétés abéliennes principalement polarisées sur un corps fini dans une classe d’isogénie ordinaire simple avec un anneau d’endomorphismes $R$ est égal soit à $0$, soit à un rapport de nombres de classes associés à $R$, à quelques petits facteurs calculables près. Cette classe d’anneaux comprend l’ordre maximal du corps CM $K$ associé à la classe d’isogénie (ce résultat était déjà connu), ainsi que l’ordre $R$ engendré sur $\mathbf{Z}$ par le Frobenius et le Verschiebung.

Pour ce dernier ordre, on peut utiliser les résultats de Louboutin pour estimer la rapport approprié des nombres de classes en fonction de la taille du corps de base et des angles de Frobenius de la classe d’isogénie. Les termes d’erreur dans nos estimations sont assez grands, mais les termes trigonométriques de l’estimée sont suggestifs : combinés avec un résultat de Vlăduţ sur la distribution des angles de Frobenius dans les classes d’isogénie, elles donnent une explication heuristique du théorème de Katz et Sarnak sur la distribution limite du multi-ensemble des angles de Frobenius pour les variétés abéliennes principalement polarisées de dimension fixée sur les corps finis.)

We show that for a large class of rings $R$, the number of principally polarized abelian varieties over a finite field in a given simple ordinary isogeny class and with endomorphism ring $R$ is equal either to $0$, or to a ratio of class numbers associated to $R$, up to some small computable factors. This class of rings includes the maximal order of the CM field $K$ associated to the isogeny class (for which the result was already known), as well as the order $R$ generated over $\mathbf{Z}$ by Frobenius and Verschiebung.

For this latter order, we can use results of Louboutin to estimate the appropriate ratio of class numbers in terms of the size of the base field and the Frobenius angles of the isogeny class. The error terms in our estimates are quite large, but the trigonometric terms in the estimate are suggestive: Combined with a result of Vlăduţ on the distribution of Frobenius angles of isogeny classes, they give a heuristic argument in support of the theorem of Katz and Sarnak on the limiting distribution of the multiset of Frobenius angles for principally polarized abelian varieties of a fixed dimension over finite fields.

Classification : 11G10, 11G15, 11G25, 14G15, 14K15, 14K22
Mots clés : Abelian variety, Frobenius eigenvalue, distribution, isogeny, complex multiplication, Katz–Sarnak

[AG17] Achter, Jeffrey D.; Gordon, Julia Elliptic curves, random matrices and orbital integrals, Pac. J. Math., Volume 286 (2017) no. 1, pp. 1-24 (with an appendix by S. Ali Altuğ) | DOI | MR | Zbl

[Bir68] Birch, Bryan J. How the number of points of an elliptic curve over a fixed prime field varies, J. Lond. Math. Soc., Volume 43 (1968), pp. 57-60 | DOI | MR | Zbl

[BL94] Buchmann, Johannes A.; Lenstra, Jr., Hendrik W. Approximating rings of integers in number fields, J. Théor. Nombres Bordeaux, Volume 6 (1994) no. 2, pp. 221-260 | DOI | MR | Zbl

[Byk97] Bykovskiĭ, Viktor A. Density theorems and the mean value of arithmetic functions in short intervals, J. Math. Sci., New York, Volume 83 (1997), pp. 720-730 translation from Russian of Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 212 (1994) 56–70. | DOI | MR

[Cox13] Cox, David A. Primes of the form $x^2+n{y}^2$: Fermat, class field theory, and complex multiplication, Pure and Applied Mathematics, John Wiley & Sons, 2013 | DOI | MR | Zbl

[Del69] Deligne, Pierre Variétés abéliennes ordinaires sur un corps fini, Invent. Math., Volume 8 (1969), pp. 238-243 | DOI | MR | Zbl

[DH98] 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 [DH00]) | DOI | MR | Zbl

[DH00] 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. 182 | DOI | MR

[Gek03] Gekeler, Ernst-Ulrich Frobenius distributions of elliptic curves over finite prime fields, Int. Math. Res. Not. (2003) no. 37, pp. 1999-2018 | DOI | MR | Zbl

[GW19] Gerhard, Jonathan; Williams, Cassandra Local heuristics and an exact formula for abelian varieties of odd prime dimension over finite fields, New York J. Math., Volume 25 (2019), pp. 123-144 | MR | Zbl

[Has19] Hasse, Helmut On the class number of Abelian number fields, Springer, 2019 (translated from the German by M. Hirabayashi, and extended with tables by M. Hirabayashi and K.-i. Yoshino) | DOI | MR | Zbl

[How95] Howe, Everett W. Principally polarized ordinary abelian varieties over finite fields, Trans. Am. Math. Soc., Volume 347 (1995) no. 7, pp. 2361-2401 | DOI | MR | Zbl

[How04] Howe, Everett W. On the non-existence of certain curves of genus two, Compos. Math., Volume 140 (2004) no. 3, pp. 581-592 | DOI | MR | Zbl

[How20] Howe, Everett W. Variations in the distribution of principally polarized abelian varieties among isogeny classes (2020) (https://arxiv.org/abs/2005.14365)

[IT20] Ionica, Sorina; Thomé, Emmanuel Isogeny graphs with maximal real multiplication, J. Number Theory, Volume 207 (2020), pp. 385-422 | DOI | MR | Zbl

[KS99] Katz, Nicholas M.; Sarnak, Peter Random matrices, Frobenius eigenvalues, and monodromy, Colloquium Publications, 45, American Mathematical Society, 1999 | DOI | MR | Zbl

[Len87] Lenstra, Jr., Hendrik W. Factoring integers with elliptic curves, Ann. Math., Volume 126 (1987) no. 3, pp. 649-673 | DOI | MR | Zbl

[Lou06] Louboutin, Stéphane R. Lower bounds for relative class numbers of imaginary abelian number fields and CM-fields, Acta Arith., Volume 121 (2006) no. 3, pp. 199-220 | DOI | MR | Zbl

[LPP02] Lenstra, Jr., Hendrik W.; Pila, Jonathan; Pomerance, Carl A hyperelliptic smoothness test. II, Proc. Lond. Math. Soc., Volume 84 (2002) no. 1, pp. 105-146 | DOI | MR | Zbl

[Mat86] Matsumura, Hideyuki Commutative ring theory, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, 1986 (translated from the Japanese by M. Reid) | MR | Zbl

[Nar04] Narkiewicz, Władysław Elementary and analytic theory of algebraic numbers, Springer Monographs in Mathematics, Springer, 2004 | DOI | MR | Zbl

[Neu99] Neukirch, Jürgen Algebraic number theory, Grundlehren der Mathematischen Wissenschaften, 322, Springer, 1999 (translated from the German by N. Schappacher) | DOI | MR | Zbl

[PL87] Picavet-L’Hermitte, Martine Ordres de Gorenstein, Ann. Sci. Univ. Blaise Pascal Clermont-Ferrand II, Volume 91 (1987) no. 24, pp. 1-32 | MR | Zbl

[Sch87] Schoof, René Nonsingular plane cubic curves over finite fields, J. Comb. Theory, Ser. A, Volume 46 (1987) no. 2, pp. 183-211 | DOI | MR | Zbl

[Shi98] Shimura, Goro Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series, 46, Princeton University Press, 1998 (revised and expanded version of [ST61]) | DOI | MR | Zbl

[ST61] Shimura, Goro; Taniyama, Yutaka Complex multiplication of abelian varieties and its applications to number theory, Publications of the Mathematical Society of Japan, 6, Mathematical Society of Japan, 1961 (revised and expanded in [Shi98]) | MR | Zbl

[Tat66] Tate, John Endomorphisms of abelian varieties over finite fields, Invent. Math., Volume 2 (1966), pp. 134-144 | DOI | MR | Zbl

[Tat71] Tate, John Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T. Honda), Séminaire Bourbaki. Vol. 1968/69: Exposés 347–363 (Lecture Notes in Mathematics), Volume 352, Springer, 1971, pp. 95-110 | DOI | MR | Zbl

[Vlă01] Vlăduţ, Serge G. Isogeny class and Frobenius root statistics for abelian varieties over finite fields, Mosc. Math. J., Volume 1 (2001) no. 1, pp. 125-139 | DOI | MR | Zbl

[Wat69] Waterhouse, William C. Abelian varieties over finite fields, Ann. Sci. Éc. Norm. Supér., Volume 2 (1969), pp. 521-560 | DOI | Numdam | MR | Zbl

[Wey97] Weyl, Hermann The classical groups: Their invariants and representations, Princeton Landmarks in Mathematics, Princeton University Press, 1997 reprint of the second edition (1946) of the 1939 original, fifteenth printing | DOI | MR | Zbl