Korean J. Math. 21 (2013), No. 4, pp. 365–374 http://dx.doi.org/10.11568/kjm.2013.21.4.365
REMARK ON AVERAGE OF CLASS NUMBERS OF FUNCTION FIELDS Hwanyup Jung Abstract. Let k = Fq (T ) be a rational function field over the finite field Fq , where q is a power of an odd prime number, and A = Fq [T ]. Let γ be a generator of F∗q . Let Hn be the subset of A consisting of monic square-free polynomials of degree n. In this paper we obtain an asymptotic formula for the mean value of L(1, χγD ) and calculate the average value of the ideal class number hγD when the average is taken over D ∈ H2g+2 .
1. Introduction and statement of result Let k = Fq (T ) be a rational function field over the finite field Fq , where q is a power of an odd prime number, and A = Fq [T ]. Let A+ be the set of monic polynomials in A and H be the subset of A+ consisting + of monic square-free polynomials. Write A+ n = {N ∈ A : deg N = n} square free D ∈ A+ , let OD and Hn = H ∩ A+ n . For any nonconstant √ be the integral closure of A in k( D) and hD be the ideal class number of OD . Hoffstein and Rosen [3] calculated the average value of the ideal class number hD when the average is taken over all monic polynomials Received July 3, 2013. Revised September 13, 2013. Accepted September 13, 2013. 2010 Mathematics Subject Classification: 11R58, 11R18, 11R29. Key words and phrases: Mean Values of L-functions, finite fields, function fields, class numbers. This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2010-0008139). c The Kangwon-Kyungki Mathematical Society, 2013.
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D of a fixed odd degree. Andrade [1] obtained an asymptotic formula for the mean value of L(1, χD ) and calculated the average value of the ideal class number hD when the average is taken over D ∈ H2g+1 . We remark that Andrade assumed that q ≡ 1 mod 4 for simplicity, but his results hold true for any odd q > 3. In a recent paper, the author [4] obtained an asymptotic formula for the mean value of L(1, χD ) and calculated the average value of the ideal class number hD when the average is taken over D ∈ H2g+2 . Note ∈ H2g+1 , the infinite place ∞k = (1/T ) of √ that if D √ k ramifies in k( D), i.e., k( D)/k is a (ramified) √ imaginary √ quadratic extension, and if D ∈ H2g+2 , ∞k splits in k( D), i.e., k( D)/k is a real quadratic extension. Let γ be a fixed generator of F∗q . Any inert imaginary quadratic extension K of k√ (i.e., ∞k is inert in K) can be written uniquely in the form K = k( γD) for some D ∈ H2g+2 . The aim of this paper is to study the asymptotic formula for the mean value of L(1, χγD ) and calculate the average value of the ideal class number hγD when the average is taken over D ∈ H2g+2 . We state our main results. Theorem 1.1. We have X L(1, χγD ) = |D|P(2) + O (2g q g ) , D∈H2g+2
where |D| = q
2g+2
and
P(s) =
Y
1−
P ∈A+
1 (1 + |P |)|P |s
.
irreducible
Since ]H2g+2 = (q − 1)q 2g+1 (see (2.1)), as a corollary of the Theorem 1.1, we have the following. Corollary 1.2. We have X 1 ]H2g+2 D∈H
L(1, χγD ) ∼ |D|P(2)
2g+2
as g → ∞. For any D ∈ H2g+2 , we have the following class number formula (see [3, Theorem 0.6]): (1.1)
q+1 qζA (2) p hγD . L(1, χγD ) = p hγD = 2 |D| 2ζA (3) |D|
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By Corollary 1.2 and the class number formula (1.1), we have the following asymptotic formula for the average of the class number hγD . Theorem 1.3. We have X p 2ζA (3)P(2) 1 hγD ∼ |D| |D| ]H2g+2 D∈H qζA (2) 2g+2
as g → ∞.
2. Preliminaries 2.1. Quadratic Dirichlet L-function. Let A+ be the set of all monic + polynomials in A and A+ n = {N ∈ A : deg N = n} (n ≥ 0). The zeta function ζA (s) of A is defined by the infinite series X ζA (s) = |N |−s . N ∈A+
It is straightforward to see that ζA (s) = 1−q11−s . For any square-free D ∈ A, the quadratic character χD is defined by the Jacobi symbol D χD (N ) = ( N ) and the quadratic Dirichlet L-function L(s, χD ) associated to χD is X L(s, χD ) = χD (N )|N |−s . N ∈A+
P∞
P We can write L(s, χD ) = n=0 σn (D)q −ns with σn (D) = N ∈A+n χD (N ). Since σn (D) = 0 for n ≥ deg D, L(s, χD ) is a polynomial in q −s of degree ≤ deg D − 1. Putting u = q −s , write deg D−1
L(u, χD ) =
X
σn (D)un = L(s, χD ).
n=0
The cardinality of Hn is #H1 = q and #Hn = (1 − q −1 )q d (n ≥ 2). In particular, we have (2.1)
#H2g+2 = (q − 1)q 2g+1 =
q 2g+2 . ζA (2)
¯ = γD for any D ∈ H2g+2 . Since ( γ ) = Fix a generator γ of F∗q . Write D N ¯ D D ¯ = (−1)n σn (D). (−1)deg N , we have ( N ) = (−1)deg N ( N ). Hence, σn (D)
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For D ∈ H2g+2 , L(u, χD¯ ) has a trivial zero at u = −1. The complete ˜ χD¯ ) is defined by L-function L(u, ˜ χD¯ ) = (1 + u)−1 L(u, χD¯ ). L(u, It is a polynomial of even degree 2g and satisfies the functional equation −1 ˜ χD¯ ) = (qu2 )g L((qu) ˜ L(u, , χD¯ ).
(2.2)
Lemma 2.1. Let χD¯ be a quadratic character, where D ∈ H2g+2 . Then −1
L(q , χD¯ ) =
g X
X
n −n
(−1) q
n=0
g −(g+1)
χD (N ) + (−1) q
+ (1 + q )q
−g
g−1 X (−1)n + (−1)g+1 X n=0
χD (N )
n=0 N ∈A+ n
N ∈A+ n −1
g X X
2
χD (N ).
N ∈A+ n
¯ n ˜ χD¯ ) = P2g σ Proof. Write L(u, ¯ ) = (1 + n=0 ˜n (D)u . Since L(u, χD ˜ ¯ ¯ ¯ ¯ ¯ (1 ≤ u)L(u, χD¯ ), we have σ0 (D) = σ ˜0 (D), σn (D) = σ ˜n−1 (D) + σ ˜n (D) ¯ ¯ n ≤ 2g) and σ2g+1 (D) = σ ˜2g (D), or (2.3)
¯ = σ ˜n (D)
n X
¯ (0 ≤ n ≤ 2g). (−1)n−i σi (D)
i=0
¯ n ˜ χD¯ ) = P2g σ By substituting L(u, n=0 ˜n (D)u into (2.2) and equating co¯ = σ ¯ −g+n or σ ¯ = σ ¯ g−n . efficients, we have σ ˜n (D) ˜2g−n (D)q ˜2g−n (D) ˜n (D)q Hence, ˜ χD¯ ) = L(u,
g X
¯ n + q g u2g σ ˜n (D)u
n=0
g−1 X
¯ −n u−n . σ ˜n (D)q
n=0
In particular, we have (2.4)
˜ −1 , χD¯ ) = L(q
g X n=0
¯ −n + q −g σ ˜n (D)q
g−1 X n=0
¯ σ ˜n (D).
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369
¯ = (−1)n σn (D), we have By substituting (2.3) into (2.4) and using σn (D) ˜ −1 , χD¯ ) = L(q
g g X 1 (−1)g q −(g+1) X n −n (−1) q σ (D) + σn (D) n 1 + q −1 n=0 1 + q −1 n=0 g−1 X (−1)n + (−1)g+1 −g +q σn (D). 2 n=0
˜ −1 , χD¯ ). So we get the result since L(q −1 , χD¯ ) = (1 + q −1 )L(q 2.2. Contribution of square parts. The square part contributions in the summation of L(1, χD¯ ) over D ∈ H2g+2 are given as follow. Proposition 2.2. We have g
(2.5)
[2] X
q −2m
m=0
X
X
g
1 = |D|P(2) − q −([ 2 ]+1) |D|P(1) + O (q g ) ,
D∈H2g+2 L∈A+ m (L,D)=1
(2.6) g
(−1)g q −(g+1)
[2] X X
X
g
1 = (−1)g q −(g+1)+[ 2 ] |D|P(1) + O (gq g )
m=0 L∈A+ D∈H2g+2 m (L,D)=1
and (2.7)
(1 + q −1 )q −g
1 + (−1)g+1 2
= (q + 1)q
]−1 −g+[ g−1 2
g−1 ] [X 2 X
X
1
m=0 L∈A+ D∈H2g+2 m (L,D)=1
1 + (−1)g+1 2
|D|P(1) + O (gq g ) .
Proof. The proofs are mild modifications of those of Proposition 3.7 in [4]. We only give the proof of (2.7). By using the fact that (see [2, Proposition 5.2])) X p Φ(L) |D| Y −1 −1 1= (1 + |P | ) + O |D| , ζA (2) |L| D∈H 2g+2
(D,L)=1
P |L
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we have
(1 + q −1 )q −g
1 + (−1)g+1 2
g−1 ] [X 2 X
m=0
g+1
L∈A+ m
X
1
D∈H2g+2 (L,D)=1 ] [ g−1 2
|D| X X Y (1 + |P |−1 )−1 ζA (2) m=0 L∈A+ m P |L ] [ g−1 2 X X p Φ(L) , |D| + O q −g |L| + m=0
= (1 + q −1 )q −g
1 + (−1) 2
L∈Am
where Φ(L) is the Euler totient function. Using the fact that = (1 − q −1 )q 2m (see [5, Proposition 2.7]), we have
L∈A+ m
g−1
] [ g−1 2
q −g
P
[ 2 ] X X p Φ(L) X X =q q −m |D| Φ(L) |L| + + m=0 m=0 L∈Am
L∈Am
[ g−1 ] 2
= (q − 1)
X
qm q
g+1 2
.
m=0
By using the fact that ( [2, Lemma 5.7])
X Y (1 + |P |−1 )−1 = q m L∈A+ m P |L
X M ∈A+ deg M ≤m
µ(M ) Y 1 , |M | 1 + |P | P |M
Φ(L)
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we have (2.8) g−1
(1 + q −1 )q −g
1 + (−1)g+1 2
[ 2 ] X Y |D| X (1 + |P |−1 )−1 ζA (2) m=0 + L∈Am P |L g−1
= (1 + q −1 )q −g
1 + (−1)g+1 2
[ 2 ] |D| X qm ζA (2) m=0 X µ(M ) Y M ∈A+ deg M ≤m
= (q + 1)q
]−1 −g+[ g−1 2
− (q + 1)q
−(g+2)
1 + (−1)g+1 2
1 + (−1)g+1 2
|D|
|M |
X M ∈A+ deg M ≤[ g−1 ] 2
|D|
X
P |M
1 1 + |P |
µ(M ) Y 1 |M | 1 + |P |
µ(M )
M ∈A+ deg M ≤[ g−1 ] 2
P |M
Y P |M
1 . 1 + |P |
Finally, by using ( [4, Lemma 3.3, Lemma 3.5]) X M ∈A+ ] deg M ≤[ g−1 2
g−1 µ(M ) Y 1 = P(1) + O q − 2 |M | 1 + |P | P |M
and X M ∈A+ ] deg M ≤[ g−1 2
µ(M )
Y P |M
1 g+1 ≤ , 1 + |P | 2
we have [ g−1 ] 2 g+1 X X Y 1 + (−1) |D| −1 −g (1 + q )q (1 + |P |−1 )−1 2 ζA (2) m=0 L∈A+ m P |L g+1 g−1 1 + (−1) |D|P(1) + O (gq g ) . = (q + 1)q −g+[ 2 ]−1 2
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2.3. Contribution of non square parts. The non square part contributions in the summation of L(1, χD¯ ) over D ∈ H2g+2 are given as follow. Proposition 2.3. We have g X X X (2.9) (−1)n q −n χD (N ) = O (2g q g ) , D∈H2g+2 n=0
N ∈A+ n N 6=
X
g −(g+1)
(2.10)
(−1) q
g X X
χD (N ) = O (2g q g )
D∈H2g+2 n=0 N ∈A+ n N 6=
and (2.11) −1
(1 + q )q
g−1 X (−1)n + (−1)g+1 X
X
−g
2
D∈H2g+2 n=0
χD (N ) = O (2g q g ) .
N ∈A+ n N 6=
Proof. As in [2, Lemma 6.4], for any non-square N ∈ A+ , we have X (2.12) χD (N ) q g+1 2deg N −1 . D∈H2g+2
Using (2.12), we have X
g g X X X X n −n (−1) q χD (N ) q −n q g+1 2n−1
D∈H2g+2 n=0
n=0
N ∈A+ n N 6=
qg
N ∈A+ n
g X
2n 2g q g ,
n=0
g −(g+1)
(−1) q
X
g X X
χD (N ) q
−(g+1)
D∈H2g+2 n=0 N ∈A+ n N 6=
g X X n=0 N ∈A+ n
g X n=0
2n q n 2g q g
q g+1 2n−1
Remark on average of class numbers of function fields
373
and −1
(1 + q )q
X
−g
g−1 X (−1)n + (−1)g+1 X 2
D∈H2g+2 n=0
χD (N )
N ∈A+ n N 6=
g−1
(1 + q −1 )q −g
X X
q g+1 2n−1
n=0 N ∈A+ n
(q + 1)
g−1 X
2n q n 2g q g .
n=0
3. Proof of Theorem 1.1 By Lemma 2.1, we have (3.1) X
X
L(1, χD¯ ) =
g X
(−1)n q −n
D∈H2g+2 n=0
D∈H2g+2
X
χD (N ) + (−1)g q −(g+1)
N ∈A+ n
g
X
X X
D∈H2g+2 n=0 −1
+ (1 + q )q
X
−g
χD (N )
N ∈A+ n
g−1 X (−1)n + (−1)g+1 X 2
D∈H2g+2 n=0
χD (N ).
N ∈A+ n
By (2.5) and (2.9), we have (3.2)
X
g X X (−1)n q −n χD (N )
D∈H2g+2 n=0
N ∈A+ n g
= |D|P(2) − q −([ 2 ]+1) |D|P(1) + O 2g+1 q g+1
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H. Jung
and, by (2.6) and (2.10), we have g X X X g −(g+1) (3.3) χD (N ) (−1) q D∈H2g+2 n=0 N ∈A+ n
g = (−1)g q −(g+1)+[ 2 ] |D|P(1) + O 2g+1 q g+1 . Similarly, by (2.7) and (2.11), we have g−1 X X (−1)n + (−1)g+1 X −1 −g (3.4) χD (N ) (1 + q )q 2 + D∈H2g+2 n=0 N ∈An g+1 g−1 1 + (−1) = (q + 1)q −g+[ 2 ]−1 |D|P(1) + O (2g q g ) . 2 It is easy to see that g 1 + (−1)g+1 g −(g+1)+[ g2 ] −g+[ g−1 ]−1 2 (−1) q + (q + 1)q − q −([ 2 ]+1) = 0. 2 Hence, by inserting (3.2), (3.3) and (3.4) into (3.1), we get X L(1, χD¯ ) = |D|P(2) + O (2g q g ) . D∈H2g+2
References [1] J.C. Andrade, A note on the mean value of L-functions in function fields. Int. J. Number Theory, 08 (2012), no. 12, 1725–1740. [2] J.C. Andrade and J.P. Keating, The mean value of L( 21 , χ) in the hyperelliptic ensemble. J. Number Theory 132 (2012), no. 12, 2793–2816. [3] J. Hoffstein and M. Rosen, Average values of L-series in function fields. J. Reine Angew. Math. 426 (1992), 117–150. [4] H. Jung, A note on the mean value of L(1, χ) in the hyperelliptic ensemble. To appear in Int. J. Number Theory. [5] M. Rosen, Number theory in function fields. Graduate Texts in Mathematics 210, Springer-Verlag, New York, 2002.
Department of Mathematics Education Chungbuk National University Cheongju 361-763, Korea E-mail :
[email protected] 
