Average values of L-functions in even characteristic

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by  Sunghan Bae

AVERAGE VALUES OF L-FUNCTIONS IN EVEN CHARACTERISTIC
SUNGHANBAE ANDHWANYUPJUNG
Abstract. Let k =Fq(T) be the rational function field over a finite field Fq, where q is
a power of 2. In this paper we solve the problem of averaging the quadratic L–functions
L(s,χu) over fundamental discriminants. Any separable quadratic extension K of k is of
theformK =k(xu),wherexu isazeroofX2+X+u=0forsomeu∈k. Wecharacterize
the family I (resp. F, F′) of rational functions u∈k such that any separable quadratic
7
1 extensionK ofkinwhichtheinfiniteprime∞=(1/T)ofkramifies(resp. splits,isinert)
0 can be written as K =k(xu) with a unique u∈I (resp. u∈F, u∈F′). For almost all
2 s∈C with Re(s)≥ 12, we obtain the asymptotic formulas for the summation of L(s,χu)
r over all k(xu) with u∈I, all k(xu) with u∈F or all k(xu) with u∈F′ of given genus.
a As applications, we obtain the asymptotic mean value formulas of L-functions at s = 1
2
M and s = 1 and the asymptotic mean value formulas of the class number hu or the class
numbertimes regulator huRu.
2

]
T
1. Introduction
N
. In Disquisitiones Arithmeticae [7], Gauss presented two famous conjectures concerning
h
t the average values of class numbers associated with binary quadratic forms over Z, which
a
m can be restated as the average values of class numbers of orders in quadratic number fields.
The imaginary case of these conjecture was first proved by Lipschitz, and the real case by
[
Siegel [17]. By the Dirichlet’s class number formula, these two conjectures can be stated

2
as averages of the values of quadratic Dirichlet L-functions at s= 1. In [17], Siegel showed
v
3 how to average over all discriminants. Let χd be the quadratic character defined by the
9 Kronecker symbol χ (n) = (d) and
d n
4
∞
1
0 L(s,χd)= χd(n)n−s
.
1 nX=1
0 be the Dirichlet series associated to χ . Siegel [17] has obtained averaging formulas for
d
7
L(1,χ ) over all positive discriminants d between 1 and N, or all negative discriminants
1 d
d such that 1 < d N. From these averaging formulas with Dirichlet’s class number
:
v | | ≤
formula, Siegelhasobtained averaging formulasfortheclass numberh or theclass number
i d
X times regulator h R over all positive discriminants d between 1 and N, or all negative
d d
r discriminants d such that 1 < d N. At the critical point s = 1, Jutila [14] derived an
a | | ≤ 2
asymptotic formula for
Lk(1,χ ) (k = 1,2),
2 d
d
X
where d runs over fundamental discriminants in the interval 0 < d X. In [8], using the
≤
Eisenstein series of 1-integral weight, for s C with Re(s) 1, Goldfeld and Hoffstein
2 ∈ ≥
obtained an asymptotic formula for L(s,χ ), where the sum is either over positive
m
square-free m between 1 and N, or over negative square-free m with 1 < m N. Putting
P | |≤
s = 1 and using the Dirichlet’s class number formula, one can average the class number h
m
or the class number times regulator h R over the fields Q(√m).
m m
Now,weintroducetheanalogousresultsinfunctionfieldsoverfinitefields. Letk = F (T)
q
betherationalfunctionfieldoverafinitefieldF ofq elements, andA = F [T]betheringof
q q
2000 Mathematics Subject Classification. 11R11, 11R29, 11R42, 11R58.
Key words and phrases. L-functions, Class numbers,Quadratic function fields.
1
2 S.BAEANDH.JUNG
polynomials. First, we consider the case of q being odd. Let χ be the quadratic character
N
defined by the Kronecker symbol χ (f)= (N) and
N f
L(s,χ )= χ (f)f −s
N N
| |
f∈A
X
f:monic
be the quadratic Dirichlet L-function associated to χ . In [10], Hoffstein and Rosen has
N
obtained a result on averaging L(1,χ ) over all non-square monic polynomials N A
N
∈
(all discriminants) of given degree. Using the averaging of L(1,χ ) with the Dirichlet’s
N
class number formula, they solved the problem of averaging the class number h or the
N
class number times regulator h R over all non-square monic polynomials N A of given
N N
∈
degree. Moreover, by definingandanalyzing metaplectic Eisenstein series in functionfields,
Hoffstein andRosen [10]also succeed inaveraging L(s,χ ) over all square-free polynomials
N
N (fundamentaldiscriminants)ofgivendegree. AndradeandKeating[2]andJung[12]have
obtained asymptotic formulas for L(1,χ ), where the sum is over all monic square-free
2 N
N of given degree, by elementary analytic method using approximate functional equation.
P
Recently, Andrade[1]andJung[13]alsohaveobtainedasymptoticformulasfor L(1,χ ).
N
In the case of q being even, Chen [5] obtained formulas of average values of L-functions
P
associated to orders in quadratic function fields, and then derived formulas of average class
numbersof theseorders. Theaim of this paperis to solve theproblemof averaging L(s,χ )
u
over fundamental discriminants in even characteristic. Any separable quadratic extension
K of k is of the form K = k(x ), where x is a zero of X2+X+u= 0 for some u k. We
u u
∈
characterize three families , ′ and of rational functions u k such that any separable
F F I ∈
quadratic extension K of k can be written uniquely as K = K(x ), where u , u ′
u
∈ F ∈ F
or u according as the infinite prime = (1/T) of k splits, is inert or is ramified in K
∈ I ∞
(see 2.2). By extending the analytic methods in [1–4,12,13,16], for almost all s C with
§ ∈
Re(s) 1, we obtain the asymptotic formulas for the summation of L(s,χ ) over all K
≥ 2 u u
with u , all K with u ′ or all K with u of given genus (see Theorem 2.4).
u u
∈ F ∈ F ∈ I
In [1,2,12,13], the authors have obtained the asymptotic formulas for the summation of
L(s,χ ) only at s= 1 and s = 1. The asymptotic formulas for the summation of L(s,χ )
N 2 u
obtained in this paper hold for almost all s C with Re(s) 1. This can be regarded as
∈ ≥ 2
an analogue of the result of Hoffstein and Rosen [10] in even characteristic. The method
of proving “Approximate” functional equations of L-functions (Lemma 3.1) in this paper
also holds in odd characteristic case and all results in 3.2 hold in odd characteristic too.
§
Thus our method can also be applied to obtain the asymptotic formulas for the summation
of L(s,χ ) for almost all s C with Re(s) 1 in odd characteristic. The methods and
N ∈ ≥ 2
calculations in [1,2,10] also can give the same asymptotic formulas for Re(s) 1. As
≥ 2
applications, we obtain the asymptotic mean value formulas of L-functions at s = 1 and
2
s = 1 (see Corollaries 2.6 and 2.7), and using the Dirichlet’s class number formula, we also
obtain the asymptotic mean value formulas of the class number h or the class number
u
times regulator h R (see Theorem 2.8 and Corollary 2.9).
u u
2. Statement of results
2.1. Some Background on A = F [T]. Let q be a power of 2. Let k := F (T) be the
q q
rational function field with a constant field F , = (1/T) the infinite prime of k, and
q
∞
A := F [T]. We denote by A+ the set of monic polynomials in A and by P the set of monic
q
irreducible polynomials in A. Throughout this paper, any monic irreducible polynomial
P P will be also called a “prime” polynomial. For a positive integer n we denote by
∈
A+ the set of monic polynomials in A of degree n and by P the set of monic irreducible
n n
polynomials in A of degree n. The norm f of a polynomial f A is defined to be
| | ∈
f := #(A/fA) = qdeg(f) for f = 0, and f := 0 for f = 0, where #X denotes the
| | 6 | |
cardinality of a set X. For any 0 = f A, let Φ(f) := #(A/fA)×, and let sgn(f) be
6 ∈
AVERAGE VALUES OF L-FUNCTIONS IN EVEN CHARACTERISTIC 3
the leading coefficient of f. Let ℘ : k k be the additive homomorphism defined by
→
℘(x) = x2+x.
The zeta function ζ (s) of A is defined for Re(s) > 1 to be the following infinite series:
A
−1
1 1
ζ (s):= = 1 , Re(s) > 1. (2.1)
A f s − P s
fX∈A+ | | PY∈P(cid:18) | | (cid:19)
It is well known that ζ (s) = 1 .
A 1−q1−s
The monic irreducible polynomials in A also satisfies the analogue of the Prime Number
Theorem. In other words we have the following.
Theorem 2.1 (Prime Polynomial Theorem). Let π (n) denote the number of monic
A
irreducible polynomials in A of degree n. Then, we have
n
qn q2
π (n)= #P = +O . (2.2)
A n
n n
!
2.2. Quadratic function field in even characteristic. Any separable quadratic exten-
sion K of k is of the form K = K := k(x ), where x is a zero of X2 +X +u = 0 for
u u u
some u k. We say that u,v k are equivalent if K = K . It is known that u and v
u v
∈ ∈
are equivalent if and only if u+v = ℘(w) for some w k (see [9] or [11]). Fix an element
∈
ξ F ℘(F ). Thefollowing lemmais duetoY.Li. Infact, Y.Liobtained theresultwhich
q q
∈ \
holds for any Artin-Schreier extensions of the rational function fields of any characteristic.
Lemma 2.2. Any separable quadratic extension K of k is of the form K = K , where
u
u k can be uniquely normalized to satisfy the following conditions:
∈
m ei Q n
u = ij + α T2ℓ−1+α, (2.3)
2j−1 ℓ
P
i=1 j=1 i ℓ=1
XX X
where P P are distinct, Q A with deg(Q ) < deg(P ), Q = 0, α 0,ξ , α F
i ∈ ij ∈ ij i iei 6 ∈ { } ℓ ∈ q
and α = 0 for n> 0.
n
6
Proof. Since it is difficult to find the reference for it, we will give the proof due to Y. Li.
We know that every element u k can be uniquely written as
∈
m ei Q n
u(T) = ij + α Tℓ+α,
Peij ℓ
Xi=1eXij=1 i Xℓ=1
where P P are distinct, deg(Q )< deg(P ), Q = 0 and α ,α F . We can remove the
i ∈ ij i iei 6 ℓ ∈ q
term Qij with 2e as follows. Let e = 2k and let M A with deg(M ) < deg(P )
Peij | ij ij ij ij ∈ ij i
i
such that
M2 Q modP .
ij ≡ ij i
Then we have
2
Qij Mij Mij Qij +Mi2j Mij
+ + = + .
Peij Pkij! Pkij Peij Pkij
i i i i i
Similarly, we can remove even degree term α T2ℓ and we get the desired form. Now it is
2ℓ
clear that any two u,v k which are normalized as in (2.3) are equivalent if and only if
u = v. ∈ (cid:3)
Let u k be normalized one as in (2.3). The infinite prime =(1/T) splits, is inert or
∈ ∞
ramified in K according as n = 0 and α = 0, n = 0 and α= ξ, or n > 0. Then the field K
u u
4 S.BAEANDH.JUNG
is called real, inert imaginary, or ramified imaginary, respectively. The discriminant D of
u
K is given by
u
m
D = P2ei if n = 0
u i
i=1
Y
and
m
D = P2ei (1/T)2n if n> 0,
u i ·
i=1
Y
and, by the Hurwitz genus formula ([18, Theorem III.4.12]), the genus g of K is given by
u u
1
g = deg(D ) 1. (2.4)
u u
2 −
For M A+, let r(M) := P and t(M) := M r(M). For P P, let ν be the
∈ P|M · ∈ P
normalized valuation at P, that is, ν (M) = e, where Pe M. Let be the set of monic
P
Q k B
polynomials M such that ν (M) = 0 or odd for any P P, that is, t(M) is a square, and
P
∈ C
be the set of rational functions D k such that D A,M and deg(D)< deg(M). For
M ∈ ∈ ∈B
M , let ℓ := 1(ν (M)+1) for any P M. Also we let be the set of rational functions
∈ B P 2 P | E
D of the form
M ∈C
D ℓP A
P,i
= ,
M P2i−1
P|M i=1
XX
where deg(A ) < deg(P) for any P M and for all 1 i ℓ . Note that for D ,
P,i | ≤ ≤ P M ∈ E
gcd(D,M) = 1 if and only if A = 0 for all P M. Let be the set of rational functions
P,ℓP 6 | F
D such that A = 0 for all P M and ′ := u+ξ :u . By the normalization in
M ∈E P,ℓP 6 | F { ∈ F}
(2.3), we can see that u K defines a one-to-one correspondence between (resp. ′)
u
7→ F F
and the set of all real (resp. inert imaginary) separable quadratic extensions of k. For any
positive integer s, let be the set of polynomials F(T) A of the form
s
G ∈
s
F(T) = α+ α T2i−1, where α 0,ξ ,α F and α = 0.
i i q s
∈ { } ∈ 6
i=1
X
Let := and := u + F : u ¯ and F , where ¯ = 0 . By
G s≥1Gs I { ∈ F ∈ G} F F ∪ { }
the normalization in (2.3), we can see that w K defines a one-to-one correspondence
w
S 7→
between and the set of all ramified imaginary separable quadratic extensions of k.
I
2.3. Hasse symbol and L-functions. Let P P. For any u k whose denominator is
∈ ∈
not divisible by P, the Hasse symbol [u,P) with values in F is defined by
2
0 if X2+X u (modP) is solvable in A,
[u,P) := ≡
(1 otherwise.
For N A prime to the denominator of u, write N = sgn(N) s Pei, where P P are
distinct∈and e 1, and define [u,N) to be s e [u,P ). i=1 i i ∈
i ≥ i=1 i i Q
For u k and 0= N A, we also define the quadratic symbol:
∈ 6 ∈ P
u ( 1)[u,N) if N is prime to the denominator of u,
:= −
N (0 otherwise.
n o
Thissymbolis clearly additiveinitsfirstvariable, andmultiplicative inthesecondvariable.
For the field K , we associate a character χ on A+ which is defined by χ (f) = u ,
u u u {f}
and let L(s,χ )bethe L-function associated to the character χ : for s C with Re(s) 1,
u u
∈ ≥
−1
χ (f) χ (P)
u u
L(s,χ ):= = 1 .
u f s − P s
f∈A+ | | P∈P(cid:18) | | (cid:19)
X Y
AVERAGE VALUES OF L-FUNCTIONS IN EVEN CHARACTERISTIC 5
It is well known that L(s,χ ) is a polynomial in q−s. Letting z = q−s, write (z,χ ) =
u u
L
L(s,χ ). Then, (z,χ )isapolynomialinzofdegree2g +1(1+( 1)ε(u)),whereε(u) = 1if
u L u u 2 −
K isramifiedimaginary andε(u) = 0otherwise. Alsowehave that (z,χ )hasa“trivial”
u u
L
zero at z = 1 (resp. z = 1) if and only if K is real (resp. inert imaginary), so we can
u
−
define the “completed” L–function as
(z,χ ) if K is ramified imaginary,
u u
L
∗(z,χ ) := (1 z)−1 (z,χ ) if K is real, (2.5)
u  u u
L − L
(1+z)−1 (z,χu) if Ku is inert imaginary,
L
which is a polynomial of even degree 2gu satisfying the functional equation
1
∗(z,χ ) = (qz2)gu ∗ ,χ . (2.6)
u u
L L qz
(cid:18) (cid:19)
2.4. Main results. For M , let
∈B
M˜ := P(νP(M)+1)/2 = t(M).
P|M
Y p
For positive integer n, let
:= M : deg(t(M)) = 2n , := D :M ,
Bn { ∈ B } Cn M ∈ C ∈Bn
:= , := , ′ := u+ξ : u .
En E ∩Cn Fn F ∩En Fn { (cid:8) ∈ Fn} (cid:9)
Under the above correspondence u K , (resp. ′) corresponds to the set of all real
7→ u Fn Fn
(resp. inert imaginary) separable quadratic extensions K of k with genus n 1.
u
−
Let = 0 . For any integers r 0 and s 1, let = u+F : u ,F .
0 (r,s) r s
F { } ≥ ≥ I { ∈ F ∈ G }
For any integer n 1, let be the union of all , where (r,s) runs over all pairs
n (r,s)
≥ I I
of nonnegative integers such that s > 0 and r +s = n. Then, under the correspondence
u K , corresponds to the set of all ramified imaginary separable quadratic extensions
u n
7→ I
K of k with genus n 1.
u
−
Lemma 2.3. For a positive integer n, we have # = qn, # = q2n, # = ζ (2)−1q2n
n n n A
B E F
and # =2ζ (2)−1q2n−1.
n A
I
Proof. The map A+ defined by M M˜ and the map A+ defined by N
Bn → n 7→ n → Bn 7→
N∗ := N2/r(N) are inverse to each other. Thus we have # = #A+ = qn. For each
Bn n
M , there are qn (resp. Φ(M˜)) D’s such that D (resp. D ). Thus
∈ Bn M ∈ En M ∈ Fn
# = qn # = q2n and
n n
E · B
# = Φ(M˜)= Φ(M˜) = ζ (2)−1q2n
n A
F
MX∈Bn M˜X∈A+n
by [15, Proposition 2.7]. Since # = 2ζ (2)−1qs for s 1, we have
s A
G ≥
n n
# = # = # # = 2ζ (2)−1q2n−1.
n (n−s,s) n−s s A
I I F · G
s=1 s=1
X X
(cid:3)
For any s C with Re(s) > 0, let
∈
1
P(s)= 1 .
− P s(P +1)
P (cid:18) | | | | (cid:19)
Y
For an arbitrary small ε > 0, let X be the set of complex numbers s C such that
ε
1 Re(s) < 1 and s 1 > ε, and X¯ = 1 X s C : Re(s) 1∈. For the first
2 ≤ | − 2| ε {2}∪ ε ∪{ ∈ ≥ }
moment of Dirichlet L–functions at s C with Re(s) 1, we have the following theorem.
∈ ≥ 2
Theorem 2.4. Let s = 1(1+log 2) 1.
1 2 q ≤
6 S.BAEANDH.JUNG
(1) For an arbitrary small ε > 0 and s X¯ , we have
ε
∈
L(s,χ )= 2α (s)q2g+1+ O(g2g2q(2−s)g) if Re(s)< s1,
u g 3g
(O(gq 2 ) if Re(s) s1,
u∈XIg+1 ≥
where α (1) = P(1) g+1+ 2 P′(1) , α (s) = ζA(2s)P(2s) for Re(s) 1, and, for
g 2 ζA(2) logq P g ζA(2) ≥
s X ,
∈ ε (cid:0) (cid:1)
ζ (2s)
α (s) = A P(2s) q(1−2s)(g+1)P(2 2s)
g
ζ (2) − −
A
n
+P(1) q(1−2s)(g−[g−21]) q(1−2s)([g2]+1) .
−
(2) For an arbitrary small ε > 0 an(cid:16)d δ > 0 and for s X¯ with s(cid:17)o1 > δ or s = 1,
ε
∈ | − |
we have
L(s,χ ) =β (s)q2g+2+ O(2g2q(2−s)g) if Re(s) < s1, (2.7)
u g 3g
u∈XFg+1 (O(gq 2 ) if Re(s) ≥ s1,
where β (1) = P(1) g+1+ζ (1)+ 2 P′(1) , β (s) = ζA(2s)P(2s) for Re(s) 1,
g 2 ζA(2) A 2 logq P g ζA(2) ≥
and, for Re(s) < 1(s = 1),
(cid:0) 6 2 (cid:1)
βg(s) = ζA(2s) P(2s) P(1)q(1−2s)([g2]+1)
ζ (2) −
A
n
+ζA(2−s)q(1−2s)(g+1) P(1)q(2s−1)([g−21]+1) P(2 2s)
ζA(1+s) − −
P(1)q−gs q[2g]−s+ ζA(cid:16)(2−s)q[g−21] . (cid:17)o
− ζA(1+s)
(3) For an arbitrary small ε > 0 and(cid:16)s X¯ , we have (cid:17)
ε
∈
L(s,χ ) = γ (s)q2g+2+ O(2g2q(2−s)g) if Re(s)< s1,
u g 3g
u∈XFg′+1 (O(gq 2 ) if Re(s)≥ s1,
where γ (1) = P(1) g+1+ ζA(0) + 2 P′(1) , γ (s) = ζA(2s)P(2s) for Re(s) 1,
g 2 ζA(2) ζA(1) logq P g ζA(2) ≥
2
and, for s X ,
ε (cid:0) (cid:1)
∈
γg(s) = ζA(2s) P(2s) P(1)q(1−2s)([g2]+1)
ζ (2) −
A
n
+ 11++qqs−−s1 q(1−2s)(g+1) P(1)q(2s−1)([g−21]+1)−P(2−2s)
+(−(cid:16)1)gP(1)(cid:17)q−gs q[g2]−s−(cid:16) 11++qqs−−s1 q[g−21] . (cid:17)o
(cid:16) (cid:16) (cid:17) (cid:17)
Remark 2.5. The restrictions of s 1 > ε inX and s 1 > δ inTheorem 2.4 are caused
| −2| ε | − |
by the facts that ζ (2s) and ζ (2 s) are unbounded as s 1 and s 1, respectively.
A A − → 2 →
Since # = 2ζ (2)−1q2g+1 and # = # ′ = ζ (2)−1q2g+2, we get from Theo-
Ig+1 A Fg+1 Fg+1 A
rem 2.4 the following asymptotic mean value formulas of L-functions at s = 1 and s = 1.
2
Compare with the results in [1,2,12,13].
Corollary 2.6. Assume that q >2 is fixed. As g , we have
→ ∞
1
(1) L(1,χ ) (g+1)P(1),
# 2 u ∼
g+1
I u∈XIg+1
1
(2) L(1,χ ) (g+1)P(1),
# 2 u ∼
g+1
F u∈XFg+1
AVERAGE VALUES OF L-FUNCTIONS IN EVEN CHARACTERISTIC 7
1
(3) L(1,χ ) (g+1)P(1).
# ′ 2 u ∼
Fg+1 u∈F′
Xg+1
Corollary 2.7. Assume that q >2 is fixed. As g , we have
→ ∞
1
(1) L(1,χ ) ζ (2)P(2),
u A
# ∼
g+1
I u∈XIg+1
1
(2) L(1,χ ) ζ (2)P(2),
u A
# ∼
g+1
F u∈XFg+1
1
(3) L(1,χ ) ζ (2)P(2).
# ′ u ∼ A
Fg+1 u∈F′
Xg+1
Let bethe integral closure of A in the quadratic function field K and h bethe ideal
u u u
O
class number of . If K is real, R denotes the regulator of . We have the following
u u u u
O O
formula which connects L(1,χ ) with h ([6, Theorem 5.2]):
u u
q−guh if K is ramified imaginary,
u u
L(1,χ ) = ζ (2)−1q−guh R if K is real, (2.8)
u  A u u u
12ζA(2)ζA(3)−1q−guhu if Ku is inert imaginary.
From Theorem 2.4 and (2.8), we obtain the following theorem.

Theorem 2.8. As q is fixed and g , we have
→ ∞
(1) hu = 2P(2)q3g+1 +O gq52g ,
u∈XIg+1 (cid:16) (cid:17)
(2) huRu = ζA(2)P(2)q3g+2 +O gq52g ,
u∈XFg+1 (cid:16) (cid:17)
(3) hu = 2ζA(2)−1ζA(3)P(2)q3g+2 +O gq52g .
u∈XFg′+1 (cid:16) (cid:17)
Corollary 2.9. As q is fixed and g , we have
→ ∞
1
(1) h ζ (2)P(2)qg,
u A
# ∼
g+1
I u∈XIg+1
1
(2) h R ζ (2)2P(2)qg,
u u A
# ∼
g+1
F u∈XFg+1
1
(3) h 2ζ (3)P(2)qg.
# ′ u ∼ A
Fg+1 u∈F′
Xg+1
3. Preliminaries
3.1. “Approximate” functional equations of L–functions. Letu k beanormalized
∈
one as in (2.3) and write
du
L(s,χ ) = A (n)q−sn with A (n)= χ (f),
u u u u
Xn=0 fX∈A+n
where d = 2g + 1(1+( 1)ε(u)). In this subsection we prove the following lemma, which
u u 2 −
is a generalization of Lemmas 2.1 in [1,12,13]. We remark that the proof of Lemma 3.1
also can be applied to obtain “Approximate” functional equations of L–functions L(s,χ )
N
in odd characteristic.
Lemma 3.1. Let s C with Re(s) 1.
∈ ≥ 2
8 S.BAEANDH.JUNG
(1) For u , we have
∈ I
gu gu−1
L(s,χ ) = A (n)q−sn+q(1−2s)gu A (n)q(s−1)n. (3.1)
u u u
n=0 n=0
X X
(2) For u , we have
∈ F
gu gu
L(s,χ ) = A (n)q−sn q−(gu+1)s A (n)+H (s), (3.2)
u u u u
−
n=0 n=0
X X
gu−1
where H (1) := ζ (2)−1q−gu (g n)A (n) and, for s = 1,
u A u u
− 6
n=0
X
gu−1 gu−1
H (s):= q(1−2s)guη(s) q(s−1)nA (n) q−sguη(s) A (n)
u u u
−
n=0 n=0
X X
with η(s) = ζA(2−s).
ζA(1+s)
(3) For u ′, we have
∈ F
gu gu
L(s,χ )= A (n)q−sn+q−(gu+1)s ( 1)n+guA (n)
u u u
−
n=0 n=0
X X
gu−1 gu−1
+ν(s)q(1−2s)gu A (n)q(s−1)n +ν(s)q−sgu ( 1)n+gu+1A (n) (3.3)
u u
−
n=0 n=0
X X
with ν(s)= 1+q−s .
1+qs−1
Proof. Write
du 2gu
(z,χ )= A (n)zn and ∗(z,χ ) = A∗(n)zn.
L u u L u u
n=0 n=0
X X
By definition (2.5), we have
A (n) if u ,
u
∈ I
A∗(n) = n A (i) if u , (3.4)
u  i=0 u ∈ F
Pni=0(−1)n−iAu(i) if u ∈ F′.
From the functional equation (2.6),Pwe have
2gu 2gu 2gu
A∗(n)zn = A∗(n)qgu−nz2gu−n = A∗(2g n)qn−guzn,
u u u u −
n=0 n=0 n=0
X X X
and equating coefficients, we have
A∗(n) = A∗(2g n)qn−gu or A∗(2g n) =A∗(n)qgu−n.
u u u − u u − u
Hence, we can write ∗(z,χ ) as
u
L
gu gu−1
∗(z,χ ) = A∗(n)zn +qguz2gu A∗(n)q−nz−n. (3.5)
L u u u
n=0 n=0
X X
AVERAGE VALUES OF L-FUNCTIONS IN EVEN CHARACTERISTIC 9
Ifu ,since (u,χ ) = ∗(u,χ ),(3.1)followsfrom(3.5)immediatelybylettingz = q−s.
u u
∈ I L L
Suppose that u . From (3.4) and (3.5), we have
∈ F
gu n gu−1 n
∗(z,χ )= A (i) zn+qguz2gu A (j) q−nz−n
u u u
L  
!
n=0 i=0 n=0 j=0
X X X X
gu zn zgu+1  
= − A (n)+H∗(z), (3.6)
u
1 z
n=0(cid:18) − (cid:19)
X
gu−1
where H∗(q−1) := q−gu (g n)A (n) and, for z = q−1,
u u
− 6
n=0
X
qguz2gu gu−1 zgu gu−1
H∗(z) := q−nz−nA (n) A (n).
1 q−1z−1 u − 1 q−1z−1 u
− n=0 − n=0
X X
Then, by multiplying (1 z) on (3.6) and putting z = q−s, we get (3.2). Finally, consider
−
the case that u ′. By (3.4) and (3.5), we have
∈ F
gu n gu−1 n
∗(z,χ )= ( 1)n−iA (i) zn+qguz2gu ( 1)n−jA (j) q−nz−n
u u u
L −  − 
!
n=0 i=0 n=0 j=0
X X X X
1 gu zgu+1 gu  
= A (n)zn + ( 1)n+guA (n)
u u
1+z 1+z −
n=0 n=0
X X
qguz2gu gu−1 zgu gu−1
+ A (n)q−nz−n+ ( 1)n+gu+1A (n). (3.7)
1+q−1z−1 u 1+q−1z−1 − u
n=0 n=0
X X
By multiplying (1+z) on (3.7) and putting z = q−s, we get (3.3). (cid:3)
3.2. Some auxiliary lemmas. All results in this subsection hold in arbitrary character-
istic. Thus we assume that q is a power of any prime number.
The following lemma is a minor modification of Theorem 17.1 in [15].
Lemma 3.2. Let ρ : A+ C and let ζ (s) be the corresponding Dirichlet series. Suppose
ρ
→
this series converges absolutely in the region Re(s) > 1 and is holomorphic in the region
s B : Re(s) = 1 except for a simple pole of at s = 1 with residue α, where
{ ∈ }
πi πi
B = s C : Im(s) < .
∈ −logq ≤ logq
(cid:26) (cid:27)
Then, there is a positive real number δ < 1 such that
ρ(f)= α(logq)qn+O qδn .
fX∈A+n (cid:16) (cid:17)
If ζ (s) α is holomorphic in Re(s) δ′, then the error term can be replaced by with
ρ − s−1 ≥
O(qδ′n).
Lemma 3.3. Let L A+. Given any ǫ > 0, we have
∈
1
Φ(f)= q2n (1+ P −1)−1+O q(1+ǫ)n .
ζ (2) | |
A
fX∈A+n PY|L (cid:16) (cid:17)
(f,L)=1
10 S.BAEANDH.JUNG
Proof. Let ζ (s) be the Dirichlet series associated to Φ. It is known that ([15, Chap 2.
Φ
equation (6)])
ζ (s 1)
A
ζ (s) = − . (3.8)
Φ
ζ (s)
A
Let ρ : A+ C be defined by ρ(f) = Φ(f) if (L,f) = 1 and ρ(f) = 0 otherwise, and ζ (s)
ρ
→
be the Dirichlet series associated to ρ:
∞
ζ (s):= Φ(f)f −s = 1+ Φ(Pn)P −ns . (3.9)
ρ
| | | |
!
fX∈A+ PY∈P nX=1
(f,L)=1 P∤L
Then, we have
1 P 1−s
ζ (s)= ζ (s) −| | ,
ρ Φ 1 P −s
P∈P(cid:18) −| | (cid:19)
Y
P|L
which has a simple pole at s = 2 and is holomorphic in the region Re(s) > 1. Hence,
ζ (s+1) has a simple pole at s = 1 and is holomorphic in the region Re(s) > 0. Then
ρ
ζ (s+ 1) is holomorphic in the region Re(s) ǫ except for a simple pole at s = 1 with
ρ
≥
residue
1
α = (1+ P −1)−1.
ζ (2)logq | |
A
P|L
Y
Applying Lemma 3.2 to ζ (s+1) with δ′ = ǫ, we find
ρ
1
Φ(f)= q2n (1+ P −1)−1+O q(1+ǫ)n .
ζ (2) | |
A
fX∈A+n PY|L (cid:16) (cid:17)
(f,L)=1
(cid:3)
Applying Lemma 3.3 with ǫ = 1, we have the following corollary.
2
Corollary 3.4. We have
1 µ(D) 1
Φ(f)= q2n+l +O q32n+l . (3.10)
ζ (2) D P +1
A
LX∈A+l (ffX,∈LA)=+n1 deDgX∈(DA)+≤l | | PY|D | | (cid:16) (cid:17)
Proof. By Lemma 3.3, we have
1
Φ(f)= q2n (1+ P −1)−1+O q32n+l ,
ζ (2) | |
A
LX∈A+l fX∈A+n LX∈A+l PY|L (cid:16) (cid:17)
(f,L)=1
and, by [2, Lemma 5.7],
µ(D) 1
(1+ P −1)−1 = ql .
| | D P +1
LX∈A+l PY|L deDgX∈(DA)+≤l | | PY|D | |
So we get the result. (cid:3)
Now we have the following lemma, which is a generalization of Lemmas 3.3, 3.4 and 3.5
of [13].
Lemma 3.5. Let h g 1,g . For any s C with Re(s) 1, we have
∈ { − } ∈ ≥ 2