Number of points on abelian varieties over finite fields

Borys Kadets (MIT)

Tuesday, April 28, 2020 - 11:00am to 12:00pm

Zoom

Abstract: The Weil conjectures for abelian varieties (proved by Weil) imply for any abelian variety $A/\mathbb{F}_q$ of dimension $g$ the inequalities $(\sqrt{q}-1)^{2g} \leqslant |A(\mathbb{F}_q)| \leqslant (\sqrt{q}+1)^{2g}$ . I will describe proofs of improved versions of these inequalities. For example, the lower bound $(\sqrt{q}-1)^{2g}$ is vacuous for $q=2,3,4$ ; I will prove that all but finitely many simple abelian varieties satisfy $A(\mathbb{F}_3) \geqslant 1.359^g$ , $A(\mathbb{F}_4) \geqslant 2.275^g$ . For $q=2$ , on the other hand, a theorem of Madan and Sät states that there are infinitely many simple abelian varieties over $\mathbb{F}_2$ with exactly one point.