On relations among Dirichlet series whose coefficients are class numbers of binary cubic forms
aa r X i v : . [ m a t h . N T ] N ov ON RELATIONS AMONG DIRICHLET SERIESWHOSE COEFFICIENTS ARECLASS NUMBERS OF BINARY CUBIC FORMS
YASUO OHNO, TAKASHI TANIGUCHI, AND SATOSHI WAKATSUKI
Abstract.
We study the class numbers of integral binary cubic forms. For eachSL ( Z ) invariant lattice L , Shintani introduced Dirichlet series whose coefficients arethe class numbers of binary cubic forms in L . We classify the invariant lattices, andinvestigate explicit relationships between Dirichlet series associated with those lattices.We also study the analytic properties of the Dirichlet series, and rewrite the functionalequation in a self dual form using the explicit relationship. Introduction
Study of the class numbers of integral binary cubic forms was initiated by G. Eisensteinand developed by many mathematicians including C. Hermite, F. Arndt, H. Davenportand T. Shintani. Davenport [D] obtained asymptotic formulas for the sum of the classnumbers of integral irreducible binary cubic forms of positive and negative discriminants.Shintani [S2] improved the error term by using the Dirichlet series whose coefficients arethe class numbers of binary cubic forms introduced in [S1].Let V Q be the space of binary cubic forms over the rational number field Q ; V Q = { x ( u, v ) = x u + x u v + x uv + x v | x , . . . , x ∈ Q } . For x ∈ V Q , the discriminant P ( x ) is defined by P ( x ) = x x + 18 x x x x − x x − x x − x x . The group Γ = SL ( Z ) acts on V Q by the linear change of variablesand P ( x ) is invariant under the action. Let L be a Γ-invariant lattice in V Q . We put L ± = { x ∈ L | ± P ( x ) > } . For x ∈ L , let Γ x be the stabilizer of x in Γ and Γ x itsorder. Definition 1.1.
For each invariant lattice L and sign ± , we put˜ ξ ± ( L, s ) := X x ∈ Γ \ L ± ( Γ x ) − | P ( x ) | s . This Dirichlet series was introduced by Shintani [S1] as an example of the zeta func-tions of prehomogeneous vector spaces. It is shown that this Dirichlet series has num-ber of curious properties such as analytic continuation or functional equation. Hetreated when the invariant lattice is either L = { x ∈ V Q | x , x , x , x ∈ Z } or L = { x ∈ V Q | x , x ∈ Z , x , x ∈ Z } , but the proof works for a general invari-ant lattice as we confirm in this paper. Note that L and L are the dual lattice to eachother with respect to the alternating form h x, y i = x y − − x y + 3 − x y − x y on V Q . Date : November 3, 2018.The first author is supported by JSPS Grant-in-Aid No. 18740020. The second author is supportedby Research Fellowships for Young Scientists of JSPS. The third author is supported by JSPS Grant-in-Aid No. 18840018.
In 1997, the first author [O] conjectured that there are simple relations between˜ ξ ∓ ( L , s ) and ˜ ξ ± ( L , s ). This was proved by Nakagawa [N]. Theorem 1.2 (Conjectured in [O], proved in [N]) . ˜ ξ − ( L , s ) = 3 s ˜ ξ + ( L , s ) and ˜ ξ + ( L , s ) = 3 s − ˜ ξ − ( L , s ) . The primary purpose of this paper is to classify the Γ-invariant lattices and investigatewhether there are similar formulas for those lattices. In Section 3 we prove the following.
Theorem 1.3 (Theorem 3.3) . There are kinds of Γ -invariant lattices up to scaling.If we denote these lattices by L , . . . , L as in Theorem 2.1, then for Dirichlet seriesassociated with L , . . . , L we have ˜ ξ − ( L , s ) = 3 s ˜ ξ + ( L , s ) , ˜ ξ + ( L , s ) = 3 s − ˜ ξ − ( L , s ) , ˜ ξ − ( L , s ) = 3 s ˜ ξ + ( L , s ) , ˜ ξ + ( L , s ) = 3 s − ˜ ξ − ( L , s ) . On the other hand, the Dirichlet series associated with L , . . . , L do not satisfy suchsimple relations as above. For example, ˜ ξ − ( L , s ) and s ˜ ξ + ( L , s ) do not coincide witheach other. These relations of the Dirichlet series are proved in Theorem 3.3 using Theorem 1.2.(In Section 3 we slightly modify the definition of the Dirichlet series.) It is likely thatthe relations among the Dirichlet series for L , . . . , L are somewhat more complicated.If we take the arithmetic subgroup Γ smaller, there appears more invariant lattices andit may be an interesting problem to study Dirichlet series associated with those lattices.We hope these problems to be answered in the future.Such a relation of the Dirichlet is expected to exist also for some other represen-tations. Among them for the space of pairs of ternary quadratic forms ( G, V ) =(GL × GL , (Sym Aff ) ∗ ⊗ Aff ), this problem is considerably interesting and beingstudied by several mathematicians including Bhargava and Nakagawa. We note thatthere are only 2 types of G Z -invariant lattices for this case.We explain a curious application of this theorem to the functional equation for ˜ ξ ± ( L i , s ).Let a = a = 0 and a = · · · = a = 2. Following Datskovsky and Wright [DW] we putΛ ± ( L i , s ) := 2 ( a i +1) s s/ π s Γ( s )Γ( s ∓
13 )Γ( s ∓
16 ) (cid:16) √ ξ + ( L i , s ) ± ˜ ξ − ( L i , s ) (cid:17) for each sign. Then Shintani’s functional equation between the vector valued functions( ˜ ξ + ( L i , − s ) , ˜ ξ − ( L i , − s )) and ( ˜ ξ + ( L i +1 , s ) , ˜ ξ − ( L i +1 , s )) ( i = 1 , , , ,
9) is diagonalizedand symmetrized as Λ ± ( L i , − s ) = ± a i − b i s − / Λ ± ( L i +1 , s )where b = 0, b = 1, b = 3 and b = b = 2. Let i be either 1, 7 or 9. Then Theorems1.2 and 1.3 state that Λ ± ( L i +1 , s ) = ± / − s Λ ± ( L i , s ). Since a i = b i holds also, we canwrite the functional equations above as follows. Theorem 1.4 (Theorem 4.8) . Let i be either , or . Then Λ ± ( L i , − s ) = Λ ± ( L i , s ) . A similar formula holds for i = 2 , or . INARY CUBIC FORMS 3
The case i = 1 , √ ξ + ( L i , s ) ± ˜ ξ − ( L i , s ) and also the equation iscompletely symmetric. We hope this equation might help us to know something on thereal nature of the Dirichlet series. Note that the Dirichlet series √ ξ + ( L i , s ) ± ˜ ξ − ( L i , s )does not have an Euler product for any L i (see Proposition 4.7.)This paper is organized as follows. In Section 2, we give the classification of theinvariant lattices without a proof. The proof is given in Section 5. In Section 3, westudy the explicit relationship of the Dirichlet series. In Section 4 we study the analyticproperties of the Dirichlet series. In Theorem 4.3 we give functional equations explicitlyand evaluate the residues of the poles. After that we study on the diagonalization of thefunctional equation and give a simple symmetric functional equation using the result ofSection 3. We also give in Theorem 4.9 the density of the class numbers of the lattices.In Section 6, we give a table of about first fifty coefficients of the Dirichlet series. Acknowledgments.
Dr. Noriyuki Abe wrote a good C++ program to compute thecoefficients of the Dirichlet series. The table of coefficients played an important role instudying the Dirichlet series. The authors express their deep gratitude to him. The au-thors are also grateful to Professor Tomoyoshi Ibukiyama for useful comments, especiallyon applications of our results to the functional equations.
Notations.
The standard symbols Q , R , C and Z will denote respectively the set ofrational, real and complex numbers and the rational integers. If V is a variety definedover a ring R and S is an R -algebra then V S denotes its S -rational points. The 1-dimensional affine space is denoted by Aff.2. Classification of invariant lattices
Let G be the general linear group of rank 2 and V the space of binary cubic forms; G = GL ,V = { x = x ( v , v ) = x v + x v v + x v v + x v | x i ∈ Aff } . We identify V with Aff via the map x ( x , x , x , x ). We define the action of G on V by ( gx )( v , v ) = 1det( g ) · x ( pv + rv , qv + sv ) , g = (cid:18) p qr s (cid:19) ∈ G, x ∈ V. The twist by det( g ) − is to make the representation faithful. For x ∈ V , let P ( x ) be thediscriminant; P ( x ) = x x − x x − x x + 18 x x x x − x x . Then we have P ( gx ) = (det g ) P ( x ). We put G = SL . We assume these are definedover Z .Let Γ ⊂ G Q be an arithmetic subgroup. The zeta functions of the prehomogeneousvector space ( G, V ) over Q are defined for each Γ-invariant lattice in V Q . In this paperwe consider the case Γ = G Z = SL ( Z ). To begin we need the classification of theinvariant lattices. For a lattice L in V Q and q ∈ Q × , we put qL = { qx | x ∈ L } . Thenif L is a Γ-invariant lattice, qL is Γ-invariant also. Up to such a scaling, G Z -invariantlattices are classified as follows. YASUO OHNO, TAKASHI TANIGUCHI, AND SATOSHI WAKATSUKI
Theorem 2.1.
Up to scaling, the following is a complete list of SL ( Z ) -invariant latticesin V Q : L = { ( a, b, c, d ) ∈ Z } L = { ( a, b, c, d ) ∈ Z | b, c ∈ Z } L = { ( a, b, c, d ) ∈ L | b + c ∈ Z } L = { ( a, b, c, d ) ∈ L | a, d, b + c ∈ Z } L = { ( a, b, c, d ) ∈ L | a, d, b + c ∈ Z } L = { ( a, b, c, d ) ∈ L | b + c ∈ Z } L = { ( a, b, c, d ) ∈ L | a + b + c, b + c + d ∈ Z } L = { ( a, b, c, d ) ∈ L | a + b + d, a + c + d ∈ Z } L = { ( a, b, c, d ) ∈ L | a + b + d, a + c + d ∈ Z } L = { ( a, b, c, d ) ∈ L | a + b + c, b + c + d ∈ Z } We give a proof of this theorem in Section 5. Each of L , L , L , L is a sublattice of L and is containing 2 L . The relations of inclusions and their indices are given by[ L : L ] = [ L : L ] = [ L : L ] = [ L : 2 L ] = 2 , [ L : L ] = [ L : L ] = [ L : 2 L ] = 4 .L L L L L L There are similar relations for L , . . . , L .We define the alternating form on V Q by h x, y i = x y − − x y + 3 − x y − x y .Then L i and 2 − L i +1 are the dual lattices to each other for i = 3 , , , Remark 2.2.
We immediately see that all of the lattices in Theorem 2.1 are invariantunder the action of ( ) ∈ G Z . Since the group G Z = GL ( Z ) is generated by ( ) and G Z , Theorem 2.1 also gives the list of GL ( Z )-invariant lattices.3. Relations of the Dirichlet series
In this section, we define the Dirichlet series for each lattice and study their relations.Let L + i = { x ∈ L i | P ( x ) > } and L − i = { x ∈ L i | P ( x ) < } . For x ∈ L i , we put G Z ,x = { γ ∈ SL ( Z ) | γx = x } and denote by G Z ,x its order. We note that G Z ,x iseither 1 or 3. Definition 3.1. (1) For i = 1 , , , ,
9, we put ξ ± ( L i , s ) = X x ∈ G Z \ L ± i ( G Z ,x ) − | P ( x ) | s . INARY CUBIC FORMS 5 (2) For i = 2 , , , ,
10, we put ξ ± ( L i , s ) = 3 s X x ∈ G Z \ L ± i ( G Z ,x ) − | P ( x ) | s . These Dirichlet series were introduced by Shintani [S1] as an example of the zetafunctions of prehomogeneous vector spaces. This definition in (2) differs from that in[S1] by the factor of 3 s . Note that if x ∈ L then P ( x ) is a multiple of 3 . It is knownthat these Dirichlet series converges for ℜ ( s ) >
1. The analytic properties are studiedin Section 4.In [O], the first author gave the following conjecture, and proved that if the conjectureis true then the Shintani’s functional equation has a simple symmetric form. Thisconjecture was proved by Nakagawa [N].
Theorem 3.2 (Nakagawa) . ξ − ( L , s ) = ξ + ( L , s ) , ξ + ( L , s ) = ξ − ( L , s ) . In this section, we prove the following analogous relations. The simplification andsymmetrization of Shintani’s functional equation in terms of this theorem is given inTheorem 4.8.
Theorem 3.3. ξ − ( L , s ) = ξ + ( L , s ) ,ξ − ( L , s ) = ξ + ( L , s ) , ξ + ( L , s ) = ξ − ( L , s ) , ξ + ( L , s ) = ξ − ( L , s ) . On the other side the table in Section 6 asserts that, for example, ξ − ( L , s ) and ξ + ( L , s ) do not coincide with each other. We will reduce Theorem 3.3 to Theorem 3.2.The proof is given after Proposition 3.8.To prove this theorem, we study the relation between different lattices. Let E and O be the set of even integers and odd integers, respectively; E = { n | n ∈ Z } , O = { n + 1 | n ∈ Z } . We write elements of L = Z as x = ( a, b, c, d ) in this section. Hence P ( x ) = b c + 18 abcd − ac − b d − a d . We first consider the lattices in L . We put ∆ = ac + b d − a d . Then P ( x ) = ( bc + ad ) −
4∆ + 16( abcd − a d ) . Definition 3.4.
Let L be a lattice in L . For l, N ∈ Z , N = 0, we put L ≡ l ( N ) = { x ∈ L | P ( x ) ≡ l mod N } . Proposition 3.5.
We have L = 2 L ∐ L , ≡ ,L = 2 L ∐ L , ≡ , We start with a lemma.
Lemma 3.6.
Let x = ( a, b, c, d ) ∈ L . YASUO OHNO, TAKASHI TANIGUCHI, AND SATOSHI WAKATSUKI (1) P ( x ) ≡ if and only if one of the following holds; (a) a, d ∈ E , b, c ∈ O , (b) a, d ∈ O , b + c ∈ O . (2) P ( x ) ≡ if and only if one of the following holds; (a) b, c ∈ E , a, d ∈ O , (b) b, c ∈ O , a + d ∈ O .Proof. Let P ( x ) ≡ ad + bc ∈ O and P ( x ) ≡ P ( x ) mod 8, what we should see is ∆ mod 2. Now the lemma follows from theobservations below. In the following congruence expression means modulo 2.(I) Assume a + d ∈ O . Then ad ∈ E , bc ∈ O , b, c ∈ O . Hence ∆ ≡ ac + bd ≡ a + d ≡ a + d ∈ E . If a, d ∈ O , then bc ∈ E and ∆ ≡ b + c + 1 ≡ b + c + 1.Hence either ( b, c ∈ E , ∆ ≡
1) or ( b + c ∈ O , ∆ ≡ a, d ∈ E , then bc ∈ O and hence ∆ ≡ (cid:3) Proof of Proposition 3.5.
We first show L = 2 L ∐ L , ≡ . Let x = ( a, b, c, d ) ∈ L , ≡ . Then by the lemma above we have a + b + c, b + c + d ∈ E and so x ∈ L . Hence L ⊃ L ∐ L , ≡ . We consider the reverse inclusion. Let x = ( a, b, c, d ) ∈ L . Then a + b + c, b + c + d ∈ E , and so a + d ∈ E . First assume a, d ∈ O . Then b + c ∈ O and hence x ∈ L , ≡ . Next assume a, d ∈ E . Then b + c ∈ E and hence either ( a, b, c, d ∈ E ) or( a, d ∈ E , b, c ∈ O ). This shows x ∈ L ∐ L , ≡ . Hence L ⊂ L ∐ L , ≡ .The equation L = 2 L ∐ L , ≡ is proved similarly. (cid:3) We next consider the lattices in L . Recall that for x ∈ L , P ( x ) is a multiple of 27.We put Q ( x ) = P ( x ) /
27. Then Q ( x ) ≡ P ( x ) mod 8. Definition 3.7.
Let L be a lattice in L . For l, N ∈ Z , N = 0, we put L ≡ ′ l ( N ) = { x ∈ L | Q ( x ) ≡ l mod N } . Since Q ( x ) ≡ P ( x ) mod 8, we have L ≡ l (8) = L ≡ ′ l (8) . Proposition 3.8.
We have L = 2 L ∐ L , ≡ ′ ,L = 2 L ∐ L , ≡ ′ , Proof.
The first one follows from L = 2 L ∐ L , ≡ we proved in Proposition 3.5 and L ∩ L = L , L ∩ L = 2 L , L , ≡ ∩ L = L , ≡ = L , ≡ ′ . The second one is proved similarly. (cid:3)
We now give a proof of Theorem 3.3.
Proof of Theorem 3.3.
Let { a n } be the coefficients of ξ − ( L , s ); ξ − ( L , s ) = X n ≥ a n n s . INARY CUBIC FORMS 7
Then by Proposition 3.5, ξ − ( L , s ) = 12 s ξ − ( L , s ) + X n ≥ ,n ≡ a n n s ,ξ − ( L , s ) = 12 s ξ − ( L , s ) + X n ≥ ,n ≡ a n n s . If we put ξ + ( L , s ) = P n ≥ b n /n s then similarly by Proposition 3.8 we have ξ + ( L , s ) = 12 s ξ + ( L , s ) + X n ≥ ,n ≡ b n n s ,ξ + ( L , s ) = 12 s ξ + ( L , s ) + X n ≥ ,n ≡ b n n s . Hence the first two formulas follows from ξ − ( L , s ) = ξ + ( L , s ) and a n = b n . The restsare proved similarly. (cid:3) We will give some properties on ξ ± ( L i , s ). These can be checked using the table ofthe coefficients of ξ ± ( L i , s ) given in Section 6. Proposition 3.9. (1)
The Dirichlet series ξ ± ( L i , s ) does not have an Euler product. (2) The linear relations of the twenty Dirichlet series { ξ ± ( L i , s ) } are exhausted bythat given in Theorems 3.2 and 3.3. Namely, the C -vector space spanned byDirichlet series by { ξ ± ( L i , s ) } is of dimension . Analytic properties of the Dirichlet series
In this section, we study analytic properties of ξ ± ( L i , s ). We also separate the contri-butions of irreducible binary cubic forms and reducible binary cubic forms in the residueformulas. Let V ird Z = { x ( v ) ∈ V Z | x ( v ) is irreducible over Q } and V rd Z = V Z \ V ird Z . Theyare G Z -invariant subsets. Definition 4.1. (1) For i = 1 , , , ,
9, we put ξ ird ± ( L i , s ) = X x ∈ G Z \ ( L ± i ∩ V ird Z ) ( G Z ,x ) − | P ( x ) | s , ξ rd ± ( L i , s ) = X x ∈ G Z \ ( L ± i ∩ V rd Z ) ( G Z ,x ) − | P ( x ) | s . (2) For i = 2 , , , ,
10, we put ξ ird ± ( L i , s ) = 3 s X x ∈ G Z \ ( L ± i ∩ V ird Z ) ( G Z ,x ) − | P ( x ) | s , ξ rd ± ( L i , s ) = 3 s X x ∈ G Z \ ( L ± i ∩ V rd Z ) ( G Z ,x ) − | P ( x ) | s . By definition we have ξ ± ( L i , s ) = ξ ird ± ( L i , s ) + ξ rd ± ( L i , s ). Definition 4.2.
For i = 1 , , , ,
9, we put a i = [ b L i : L i +1 ] and 2 b i = [ V Z : L i ], where b L i is the dual lattice of L i with respect to the bilinear form h x, y i .It is easy to see that ( a i , b i ) is (0 , , , , ,
2) for i = 1 , , , , YASUO OHNO, TAKASHI TANIGUCHI, AND SATOSHI WAKATSUKI
Theorem 4.3. (1)
The Dirichlet series ξ ± ( L i , s ) can be continued holomorphicallyto the whole complex plane except for simple poles at s = 1 and / . Furthermore,they satisfy the following functional equations (cid:18) ξ + ( L i , − s ) ξ − ( L i , − s ) (cid:19) = 2 a i s − b i s − π s Γ( s ) Γ( s −
16 )Γ( s + 16 ) (cid:18) sin 2 πs sin πs πs sin 2 πs (cid:19) (cid:18) ξ + ( L i +1 , s ) ξ − ( L i +1 , s ) (cid:19) where i = 1 , , , , . (2) The Dirichlet series ξ ird ± ( L i , s ) and ξ rd ± ( L i , s ) have meromorphic continuations tothe whole complex plane. The first one is holomorphic for ℜ ( s ) > / except forsimple poles at s = 1 and s = 5 / . The second one is holomorphic for ℜ ( s ) > / except for a simple pole at s = 1 . (3) Let α i, ± = Res s =1 ξ ± ( L i , s ) , β i, ± = Res s =5 / ξ ± ( L i , s ) ,α ird i, ± = Res s =1 ξ ird ± ( L i , s ) , α rd i, ± = Res s =1 ξ rd ± ( L i , s ) . Then if we put α = π , β = 3 / (2 π ) / ζ (cid:18) (cid:19) Γ (cid:18) (cid:19) Γ (cid:18) (cid:19) − , the values are given by Table 1.Proof. For L and L , Shintani [S1, S2] proved this theorem by establishing the theoryof zeta functions associated with the space of binary cubic forms and the space of binaryquadratic forms. His global theory was rewritten in the adelic language by Wright [W]and the second author [T]. We would like to mention that a quite simpler version of theglobal theory for the space of binary cubic forms [W] were given by Kogiso [K]. Let A and A f be the rings of adeles and finite adeles of Q , respectively. Note that A f = b Z ⊗ Z Q and A = A f × R , where b Z is the profinite completion of Z . Let S ( V A ), S ( V A f ) and S ( V R ) be the spaces of Schwartz-Bruhat functions on each of the indicated domains.Let Φ f ∈ S ( V A f ) be the characteristic function of L i ⊗ Z b Z ⊂ V A f and Φ ∞ ∈ S ( V R )arbitrary. Then by considering the global zeta functions in [T, W] with the test functionΦ f ⊗ Φ ∞ ∈ S ( V A ), we can prove the theorem the same way as [S1, S2]. Here we illustratethe proof of (3) with i = 3 , , ,
9. We fix a prime p . We fix any Haar measures du on Q p and d × t on Q × p . For t ∈ Q × p , we put | t | p = d ( tu ) /du . For Φ ∈ S ( V Q p ), we define A ird p (Φ) = Z Q p Φ( u , u , u , u ) du du du du , A rd p (Φ) = Z Q × p × Q p | t | p Φ(0 , t, u , u ) d × tdu du , B p (Φ) = Z Q × p × Q p | t | / p Φ( t, u , u , u ) d × tdu du du . Let Φ i be the characteristic function of L i ⊗ Z p . Since i = 3 , , , i = Φ unless p = 2. Hence by [T, Proposition 8.6], we have α ird i, ± α ird1 , ± = A ird2 (Φ i ) A ird2 (Φ ) , α rd i, ± α rd1 , ± = A rd2 (Φ i ) A rd2 (Φ ) , β i, ± β , ± = B (Φ i ) B (Φ ) . INARY CUBIC FORMS 9 i α i, + α α α α α α α α α αβ i, + β β β √ β β √ β √ √ β √ β √ β √ βα ird i, + α α α α α α α α α αα rd i, + α α α α α α α α α αα i, − α α α α α α α α α αβ i, − √ β √ β √ √ β √ β √ β β √ β β β βα ird i, − α α α α α α α α α αα rd i, − α α α α α α α α α α Table 1.
The computations of the right hand sides in the equations are easily carried out. Forexample, A ird2 (Φ ) A ird2 (Φ ) = 12 , A rd2 (Φ ) A rd2 (Φ ) = 14 , B (Φ ) B (Φ ) = 14 . Since α ird1 , ± , α rd1 , ± and β , ± are known, we obtain the value. Note that α i, ± = α ird i, ± + α rd i, ± .The rest are proved similarly and we omit the detail. Note that a = a = a = a = 2in (1) comes from the fact that for i = 3 , , , L i with respect tothe alternating form on V is 2 − L i +1 . Also b = 1, b = 3 and b = b = 2 are because[ L : L ] = 2, [ L : L ] = 8 and [ L : L ] = [ L : L ] = 4, respectively. (cid:3) Remark 4.4.
As in [O, Proposition 2.1], the functional equation in the theorem iscompatible with Theorem 3.3. For example, from ξ − ( L , s ) = ξ + ( L , s ) and Theorem4.3 (1) for i = 7, we can deduce ξ − ( L , s ) = 3 ξ + ( L , s ).We discuss on the diagonalization of the functional equation in Theorem 4.3 (1) fol-lowing [DW, Proposition 4.1] and a related important observation given in [O, p.1088].Let a i +1 = a i for i = 1 , , , , Definition 4.5.
For 1 ≤ i ≤
10 and each sign ± , we putΛ ± ( L i , s ) = 2 ( a i +1) s s/ π s Γ( s )Γ( s ∓
16 )Γ( s ∓
13 ) (cid:16) √ ξ + ( L i , s ) ± ξ − ( L i , s ) (cid:17) . As a corollary to Theorem 4.3 we have the following.
Corollary 4.6. (1)
For i = 1 , , , , , Λ ± ( L i , − s ) = ± − / a i − b i Λ ± ( L i +1 , s ) . (2) Let ≤ i ≤ . The function Λ + ( L i , s ) is holomorphic except for simple polesat s = 0 , / , / , , while Λ − ( L i , s ) is holomorphic except for simple poles at s = 0 , . (3) Let ≤ i ≤ . The set of zeros of the Dirichlet series √ ξ + ( L i , s ) + ξ − ( L i , s ) and √ ξ + ( L i , s ) − ξ − ( L i , s ) in the negative real axis are respectively given by {− n | n ∈ Z ≥ } ∪ {− n + 1 / | n ∈ Z ≥ } ∪ {− n + 11 / | n ∈ Z ≥ } , {− n | n ∈ Z ≥ } ∪ {− n + 5 / | n ∈ Z ≥ } ∪ {− n + 7 / | n ∈ Z ≥ } , where we put Z ≥ = { n ∈ Z | n ≥ } .Proof. By a simple computation we can prove that the equalities in (1) are equivalent tothe functional equation given in Theorem 4.3 (1). (2) follows from the values of residuesgiven in Theorem 4.3 (3) and equalities (1) of this corollary. (3) follows from (2) andDefinition 4.5. (cid:3)
It is interesting that the poles of Λ − ( L i , s ) at s = 5 / √ ξ + ( L i , s ) ± ξ − ( L i , s ) has an Euler product. The answer is negative. Proposition 4.7.
None of the Dirichlet series √ ξ + ( L i , s ) + ξ − ( L i , s ) , √ ξ + ( L i , s ) − ξ − ( L i , s ) (1 ≤ i ≤ has an Euler product.Proof. If a Dirichlet series P n ≥ c n /n s has an Euler product, then c c pq = c p c q for anydistinct primes p and q . We can immediately confirm that any of our Dirichlet seriesdoes not satisfy this relation for p = 3 and q = 5 using the table given in Section 6. (cid:3) Now we assume i = 1 , ,
9. Then Theorems 3.2, 3.3 assert Λ ± ( L i +1 , s ) = ±√ ± ( L i , s ).Since a i = b i also, the functional equation in Corollary 4.6 (1) turns out to be of a singlefunction Λ ± ( L i , s ). Theorem 4.8.
For i = 1 , , , , , , Λ ± ( L i , − s ) = Λ ± ( L i , s ) . Namely, for i = 1 , , , , ,
10, the function Λ ± ( L i , s ) is invariant if we replace s by1 − s .We conclude this section with deriving asymptotic behavior of some arithmetic func-tions. For n ∈ Z , n = 0, let h ( L i , n ) be the number of G Z -orbit in L i ∩ V ird Z withdiscriminant n . Applying Tauberian theorem, Shintani [S2, Theorem 4] obtained anasymptotic formula of the function P < ± n Let i be either , , , or . For any ε > , X < ± n Let i be either , , , or . For any ε > , X < ± n In this section, we prove Theorem 2.1. We use an argument similar to [IS, Section3]. Let L be a SL ( Z )-invariant lattice. By taking some constant multiple if necessary,we can assume that L is contained in L and that there exists an element x ∈ L suchthat p − x L for each prime p . Such an element x is called primitive for p . We put( L ) p = L ⊗ Z Z p for a prime p . In the following, we prove that ( L ) p = ( L ) p ( p = 2 , 3) inLemma 5.1, ( L ) = ( L ) or ( L ) in Lemma 5.2, and ( L ) = ( L ) , ( L ) , ( L ) , ( L ) or ( L ) in Lemma 5.4. It is easy to see that the lattices L , L , . . . , L are G Z -invariant.Therefore we get Theorem 2.1 by these facts, because L = ∩ p :prime ( V Q ∩ ( L ) p ).From now, we shall prove Lemmas 5.1, 5.2 and 5.4. Since SL ( Z p ) contains SL ( Z ) asa dense subgroup, ( L ) p is SL ( Z p )-invariant. We put u ( α ) = (cid:18) α (cid:19) , w = (cid:18) − (cid:19) and E = (1 , , , E = (0 , , , E = (0 , , , E = (0 , , , u ( α ) on L is given by u ( α ) · x = ( x + αx + α x + α x , x + 2 αx + 3 α x , x + 3 αx , x ) . For x ∈ L , we put ψ ( x ) = u (1) · x − x = ( x + x + x , x + 3 x , x , ∈ L. Lemma 5.1. If p = 2 , , then ( L ) p = ( L ) p .Proof. Let x = ( x , x , x , x ) ∈ L be primitive for p .First we assume that x ∈ Z × p or x ∈ Z × p . By considering the action of w , we mayassume x ∈ Z × p . Let X = x − u ( − − x − x ) · x . Then since X is of the form ( ∗ , ∗ , , − ψ ( ψ ( X )) = (1 , , , E = u ( − · (1 , , , 0) and E = ψ ( E ), wehave E , E , E , E ∈ ( L ) p . Hence ( L ) p = ( L ) p .Second we assume x , x Z × p . Then we have x ∈ Z × p or x ∈ Z × p . We may assume x ∈ Z × p . Since the first component of u (1) · x + u ( − · x − x is 2 x ∈ Z × p , by theargument above we have ( L ) p = ( L ) p . (cid:3) Lemma 5.2. ( L ) = ( L ) or ( L ) .Proof. Let x = ( x , x , x , x ) ∈ L be primitive for 3.First we assume x ∈ Z × or x ∈ Z × . Taking the action of w into account, wemay assume x ∈ Z × . Let X = (2 x + 3 x ) − ψ ( x ) = ( x ′ , , x ′ , 0) and X = (2 x +6 x ) − ψ ( ψ ( x )) = (1 , x ′ , , x ′ , x ′ ∈ Z . Further we put X = X − x ′ X =(0 , − x ′ x ′ , x ′ , − x ′ x ′ ∈ Z × , X = (1 − x ′ x ′ ) − ( w · X ) = (0 , x ′′ , , x ′′ Z × , X = u ( − − x ′′ ) · X = ( x ′′ , , , X = ψ ( X ) = (1 , , , E = 2 − ψ ( X )and E = 2 − ( X − E ), we have ( L ) = ( L ) .Second we assume x , x Z × . Then we have x ∈ Z × or x ∈ Z × . We may assume x ∈ Z × . We have X = ψ ( x ) = ( x + x + x , x + 3 x , x , x + x + x ∈ Z × ,2 x + 3 x ∈ Z , 3 x ∈ Z × , X = u ( − − x − · − (2 x + 3 x )) · X = ( x ′ , , x , x ′ ∈ Z × , 3 x ∈ Z × . Then since x − ψ ( X ) = 3 E + 6 E , 2 − ψ (3 E + 6 E ) = 3 E ,3 E = 2 − ((3 E + 6 E ) − E ) and E = x ′− · ( X − x E ), we get ( L ) ⊂ ( L ) .We see ( L ) ⊂ ( L ) ⊂ ( L ) from the above results. Suppose ( L ) = ( L ) . Since( L ) / ( L ) is represented by the set { aE + bE ; 0 ≤ a, b ≤ } , ( L ) has an element ofthe form aE + bE for some ( a, b ) = (0 , L ) = ( L ) . So we get thislemma. (cid:3) Lemma 5.3. ( L ) contains ( L ) or ( L ) .Proof. Let x = ( x , x , x , x ) ∈ L be primitive for 2.(i) We assume x ∈ Z × or x ∈ Z × . We may assume x ∈ Z × . Let X = u ( − − x − x ) · x = ( ∗ , ∗ , , x ), X = (3 x ) − ψ ( X ) = ( x ′ , , , E +2 E = ψ ( X ) and 2 E = ψ ( ψ ( X )), we have 2 E , 2 E , 2 E , 2 E ∈ ( L ) .(i-a) We assume x ′ Z × . We have E + E = X − (2 − x ′ ) · (2 E ) ∈ ( L ) . Since L = Z (2 E ) + Z (2 E ) + Z ( E + E ) + Z (2 E ), we get ( L ) ⊂ ( L ) .(i-b) We assume x ′ ∈ Z × . From x ′ = 1 + x ′′ , ( x ′′ ∈ Z ), we have X − (2 − x ′′ ) · (2 E ) = E + E + E . Since L = Z ( E + E + E ) + Z ( E + E + E ) + Z (2 E ) + Z (2 E ). weget ( L ) ⊂ ( L ) .(ii) We assume x , x Z × .(ii-a) We assume x + x ∈ Z × . Since the first component of ψ ( x ) is x + x + x ∈ Z × ,we can reduce the case (ii-a) to the case (i).(ii-b) We assume x + x Z × . Since x is primitive, we have x , x ∈ Z × . We have X =( x + 3 x ) − ψ ( ψ ( x )) = (2 , c, , c ∈ Z , X = w − · X = − cE + 2 E . Furthermorewe put X = x − (2 − x ) · X − (2 − x ) · X = (0 , α, β, α = x − − x c ∈ Z × , β = x +2 − x c ∈ Z × . Let X = ψ ( X ) − − ( α + β ) X = (0 , β − − ( α + β ) c, , β − − ( α + β ) c ∈ Z × . Hence we have 2 E = ( β − − ( α + β ) c ) − X , 2 E = X − (2 − c ) · (2 E ), 2 E , 2 E ∈ ( L ) , E + E = X − − ( α − · (2 E ) − − ( β − · (2 E ) ∈ ( L ) .Therefore we get ( L ) ⊂ ( L ) . (cid:3) Lemma 5.4. ( L ) = ( L ) , ( L ) , ( L ) , ( L ) or ( L ) .Proof. Form Lemma 5.3, we know ( L ) ⊂ ( L ) ⊂ ( L ) or ( L ) ⊂ ( L ) ⊂ ( L ) . Hencewe have only to take all representation elements of ( L ) / ( L ) , ( L ) / ( L ) and computeall cases for subspaces containing representation elements.(I) We treat the case ( L ) ⊂ ( L ) ⊂ ( L ) . Let ( L ) = ( L ) . Since L = Z E + Z E + Z ( E + E ) + Z E and L = Z (2 E ) + Z (2 E ) + Z ( E + E ) + Z (2 E ), ( L ) / ( L ) is represented by the set { aE + bE + cE ; 0 ≤ a, b, c ≤ } .(I-1) ( L ) contains one of E , E , E + E . We easily see that ( L ) contains ( L ) . Since( L ) / ( L ) ∼ = Z / Z , ( L ) is either( L ) or ( L ) .(I-2) ( L ) contains either E , E + E or E + E . Since E = ψ ( E ) = ψ ( E + E ) = ψ ( w · ( E + E )) − E , we have ( L ) = ( L ) .(I-3) ( L ) contains E + E + E . Since L = Z ( E + E + E ) + Z ( E + E + E ) + Z (2 E ) + Z (2 E ), we see ( L ) ⊂ ( L ) . Furthermore ( L ) / ( L ) is represented by { , E , E , E + E } . If ( L ) contains one of this representation element, then we have( L ) = ( L ) . Therefore ( L ) = ( L ) or ( L ) .(II) We treat the case ( L ) ⊂ ( L ) ⊂ ( L ) . Suppose ( L ) = ( L ) . Since L = Z E + Z E + Z ( E + E + E ) + Z ( E + E + E ) and L = Z ( E + E + E ) + Z ( E + E + E ) + Z (2 E ) + Z (2 E ), ( L ) / ( L ) is represented by { aE + bE ; 0 ≤ a, b ≤ } .(II-1) ( L ) contains E . We have ( L ) ⊂ ( L ) . Hence we have ( L ) = ( L ) or ( L ) .(II-2) ( L ) contains E or E + E . Since ψ ( E ) = ψ ( E + E ) = E , we have ( L ) = ( L ) . INARY CUBIC FORMS 13 Form (I) and (II), we get this lemma. (cid:3) Table of the coefficients We give the table of about first fifty coefficients of the Dirichlet series ξ ± ( L i , s ). Inthe table, we give the value multiplied by 3 for the each coefficient except for ξ + ( L i , s ), i = 2 , , , , 10 where in which cases we give the exact value of the coefficients. Hencethe table means, for example, ξ + ( L , s ) = 1 / s + 111 s + 119 s + 4 / s + 135 s + 143 s + 148 s + 151 s + . . . ,ξ − ( L , s ) = 1 / s + 19 s + 1 / s + 117 s + 125 s + 133 s + 141 s + 5 / s + . . . ,ξ + ( L , s ) = 11 s + 39 s + 116 s + 317 s + 325 s + 333 s + 341 s + 549 s + . . . .L − L +2 L − L +4 L − L +6 L − L +8 L − L +10 L +1 L − L +3 L − L +5 L − L +7 L − L +9 L − References [D] H. Davenport. On the class-number of binary cubic forms I and II. London Math. Soc. , 26:183–198, 1951. Corrigendum: ibid., 27:512, 1952.[DW] B. Datskovsky and D.J. Wright. The adelic zeta function associated with the space of binarycubic forms II: Local theory. J. Reine Angew. Math. , 367:27–75, 1986.[IS] T. Ibukiyama and H. Saito. On L -functions of ternary zero forms and exponential sums of Leeand Weintraub. J. Number Theory , 48:252–257, 1994. [K] T. Kogiso. Simple calculation of the residues of the adelic zeta function associated with the spaceof binary cubic forms. J. Number Theory , 51:233–248, 1995.[N] J. Nakagawa. On the relations among the class numbers of binary cubic forms. Invent. Math. ,134:101–138, 1998.[O] Y. Ohno. A conjecture on coincidence among the zeta functions associated with the space ofbinary cubic forms. Amer. J. Math. , 119:1083–1094, 1997.[S1] T. Shintani. On Dirichlet series whose coefficients are class-numbers of integral binary cubicforms. J. Math. Soc. Japan , 24:132–188, 1972.[S2] T. Shintani. On zeta-functions associated with vector spaces of quadratic forms. J. Fac. Sci.Univ. Tokyo, Sect IA , 22:25–66, 1975.[T] T. Taniguchi. Distributions of discriminants of cubic algebras. Preprint 2006, math.NT/0606109.[W] D.J. Wright. The adelic zeta function associated to the space of binary cubic forms part I: Globaltheory. Math. Ann. , 270:503–534, 1985. (Y. Ohno) Department of Mathematics, Kinki University, Kowakae 3-4-1, Higashi-Osaka, Osaka 577-8502, Japan/ Max-Planck-Institut f¨ur Mathematik, Vivatsgasse 7,53111 Bonn, Germany E-mail address : [email protected] (T. Taniguchi) Department of Mathematical Sciences, Faculty of Science, Ehime Uni-versity, Bunkyocho 2-5, Matsuyama-shi, Ehime, 790-8577, Japan E-mail address : [email protected] (S. Wakatsuki) Department of Mathematics, Graduate School of Science, KanazawaUniversity, Kakumamachi, Kanazawa, Ishikawa, 920-1192, Japan E-mail address ::