# On the Number of Rational Points of Jacobians over Finite Fields

(1)

## On the Number of Rational Points of Jacobians overFinite Fields

(2)

q

qf

q

X

i

X

X

−s

X

e

q

q

2g

2g

LM D

g−1

2

1

(3)

g−1

n=0

n

g−2

n=0

g−1−n

n

g

i=1

i

2

n

q

q

1

2

r

r∈D1

r

r∈D2

1

2

1

qr

2

qr

r

r∈D1

r

r

r∈D2

r

BRT

BRT

2

q

r∈D1

qr

r

r

g

r∈D2

r

r

φqr

qr

qr

r

r

qr 0

−r

mr

Φqr+mr+1

BRT

g

q

g

x1−1

x1

2

g

q

2g−1

r=2

qr

q

2

(4)

q

q

q

2

q

q

g−1

r=0

g−1−1

q

AHL

q

n

n

n

m,v

+∞

n=1

mn

mn

v

1,v

−1

+∞

f=1

qf

f,v

v

v

2g

j=1

v

j

n

n1

−1

0

1

2

3

0

N

n=1

−1

−n

m|n

qm

N

f=1

f

qf

1

N

n=1

2

N

n=1

n

3

2g

j=1

N

n=1

12

j

n

3

(5)

0

N

f=1

qf

f

f

ε0

N

f=1

qf

f

f

1

2

ε2

3

ε3

0

2

i

1

N2

2

34N

0

N

2

3

N2

1

2

2

32

32

N

n=1

n

+∞

n=N+1

n

N+1

+∞

n=0

n

N

2

3

2g

i=1

j

3

2g

j=1

N

n=1

12

j

n

2g

j=1

j

N

n=1

j

n

j

3

2g

j=1

N

j

N2

4

(6)

1−g

q

N

f=1

f

qf

N

n=1

−n

3

N

r=1

qr

Nr

f=1

rf

N

n=1

−n

3

3

3

N2

min

max

min

g

N

f=1

f

qf

N

n=1

−n

N2

max

g

N

f=1

f

qf

N

n=1

−n

N2

qf

qf

q

N

f=1

qf

f

f

0

2

3

N

n=1

N

f=1

qf

f

f

N

n=1

−n

0

3

5

(7)

i

q

i

qr

i→∞

Φqr(Xi)

gi

i→∞

i

i

r=1

qr

r

r

min

max

i

min

BRT

AHL

LZ

qr

qr

qd

qr

r

r/2

q

k

q

k∈N

q

k

k

k+1

k

k

q

0

q

0

k+1

k

k+1

k+1

k

k+1

q

r

k

qr

q

r 2

q

r 21

q

r

2

q

k

q

q

0

k

qr

k

LZ

2

3

4

k

1

k

BRT

AHL

LZ

25

25

25

k

1

k

2

k

BRT

LZ

6

(8)

2

4

AHL

2

4

2

2

3

2

LZ

k

q3

q

q−1

q(q−1)

q−1

q−1

q

k

q

q

8

k

2

k

BRT

k

1

k

BRT

LZ

13

13

30

30

k

3

k

BRT

LZ

k

q2

2

k

1

k

2

k

3

k

BRT

LZ

31

31

77

77

k

2

k

8

k

3

k

6

k

BRT

LZ

14

14

k

1

k

2

k

BRT

LZ

17

17

48

48

k

4

7

(9)

k

1

k

2

k

3

k

BRT

LZ

16

16

52

52

k

9

k

1

k

2

k

BRT

LZ

14

14

45

45

k

4

k

1

k

2

k

BRT

LZ

18

18

86

86

BRT

AHL

LZ

LZ

BRT

LZ

BRT

min

qr

LZ

qr

LZ

8

(10)

139

### Laboratoire de Mathématiques de Besançon, UFR Sciences et techniques 16, route de Gray 25 030 Besançon, France

philippe.lebacque@univ-fcomte.fr