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Number of points on abelian varieties over finite fields

Borys Kadets (MIT)
Tuesday, April 28, 2020 - 11:00am to 12:00pm

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. 

The seminar will be held on zoom.  Please use the link below to join

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