We show that for a large class of rings , the number of principally polarized abelian varieties over a finite field in a given simple ordinary isogeny class and with endomorphism ring is equal either to , or to a ratio of class numbers associated to , up to some small computable factors. This class of rings includes the maximal order of the CM field associated to the isogeny class (for which the result was already known), as well as the order generated over 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.
Nous montrons que pour une grande classe d’anneaux , le nombre de variétés abéliennes principalement polarisées sur un corps fini dans une classe d’isogénie ordinaire simple avec un anneau d’endomorphismes est égal soit à , soit à un rapport de nombres de classes associés à , à quelques petits facteurs calculables près. Cette classe d’anneaux comprend l’ordre maximal du corps CM associé à la classe d’isogénie (ce résultat était déjà connu), ainsi que l’ordre engendré sur 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.)
Accepted:
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Keywords: Abelian variety, Frobenius eigenvalue, distribution, isogeny, complex multiplication, Katz–Sarnak
@article{AHL_2022__5__677_0, author = {Howe, Everett W.}, title = {Variations in the distribution of principally polarized abelian varieties among isogeny classes}, journal = {Annales Henri Lebesgue}, pages = {677--702}, publisher = {\'ENS Rennes}, volume = {5}, year = {2022}, doi = {10.5802/ahl.133}, language = {en}, url = {http://www.numdam.org/articles/10.5802/ahl.133/} }
TY - JOUR AU - Howe, Everett W. TI - Variations in the distribution of principally polarized abelian varieties among isogeny classes JO - Annales Henri Lebesgue PY - 2022 SP - 677 EP - 702 VL - 5 PB - ÉNS Rennes UR - http://www.numdam.org/articles/10.5802/ahl.133/ DO - 10.5802/ahl.133 LA - en ID - AHL_2022__5__677_0 ER -
Howe, Everett W. Variations in the distribution of principally polarized abelian varieties among isogeny classes. Annales Henri Lebesgue, Volume 5 (2022), pp. 677-702. doi : 10.5802/ahl.133. http://www.numdam.org/articles/10.5802/ahl.133/
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