Abstract
Let A be a simple Abelian variety of dimension g over the field Fq. The paper provides improvements on the Weil estimates for the size of A(Fq). For an arbitrary value of q we prove (⌊(q-1)2⌋+1)g≤#A(Fq)≤(⌈(q+1)2⌉-1)g holds with finitely many exceptions. We compute improved bounds for various small values of q. For instance, the Weil bounds for q= 3 , 4 give a trivial estimate # A(Fq) ≥ 1 ; we prove # A(F3) ≥ 1. 359 g and # A(F4) ≥ 2. 275 g hold with finitely many exceptions. We use these results to give some estimates for the size of the rational 2-torsion subgroup A(Fq) [2] for small q. We also describe all abelian varieties over finite fields that have no new points in some finite field extension.
Original language | English |
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Pages (from-to) | 465-473 |
Number of pages | 9 |
Journal | Mathematische Zeitschrift |
Volume | 297 |
Issue number | 1-2 |
DOIs | |
State | Published - Feb 2021 |
Externally published | Yes |
Bibliographical note
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