Estimates for the number of rational points on simple abelian varieties over finite fields
Name
209_2020_2520_ReferencePDF.pdf
Size
472.07 KB
Format
Adobe PDF
Checksum (MD5)
102ad18d8226fad619b0af3601f3a073
Author(s)
Kadets, Borys
Date Issued
March 28, 2020
Publisher
Springer Berlin Heidelberg
Version
Author's final manuscript
Abstract
Abstract
Let A be a simple Abelian variety of dimension g over the field
$$\mathbb {F}_q$$
F
q
. The paper provides improvements on the Weil estimates for the size of
$$A(\mathbb {F}_q)$$
A
(
F
q
)
. For an arbitrary value of q we prove
$$(\lfloor (\sqrt{q}-1)^2 \rfloor + 1)^g \le \#A(\mathbb {F}_q) \le (\lceil (\sqrt{q}+1)^2 \rceil - 1)^{g}$$
(
⌊
(
q
-
1
)
2
⌋
+
1
)
g
≤
#
A
(
F
q
)
≤
(
⌈
(
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
$$q=3,4$$
q
=
3
,
4
give a trivial estimate
$$\#A(\mathbb {F}_q) \ge 1$$
#
A
(
F
q
)
≥
1
; we prove
$$\# A(\mathbb {F}_3) \ge 1.359^g$$
#
A
(
F
3
)
≥
1
.
359
g
and
$$\# A(\mathbb {F}_4) \ge 2.275^g$$
#
A
(
F
4
)
≥
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
$$A(\mathbb {F}_q)[2]$$
A
(
F
q
)
[
2
]
for small q. We also describe all abelian varieties over finite fields that have no new points in some finite field extension.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s00209-020-02520-w
