Estimates for the number of rational points on simple abelian varieties over finite fields

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.