On multiplier systems and theta functions of half-integral weight for the Hilbert modular group \mathrm{SL}_2(\mathfrak{o})
aa r X i v : . [ m a t h . N T ] F e b ON MULTIPLIER SYSTEMS AND THETA FUNCTIONS OFHALF-INTEGRAL WEIGHT FOR THE HILBERTMODULAR GROUP SL ( o ) HIROSHI NOGUCHI
Abstract.
Let F be a totally real number field and o the ring of inte-gers of F . We study theta functions which are Hilbert modular forms ofhalf-integral weight for the Hilbert modular group SL ( o ). We obtain anequivalent condition that there exists a multiplier system of half-integralweight for SL ( o ). We determine the condition of F that there exists atheta function which is a Hilbert modular form of half-integral weightfor SL ( o ). The theta function is defined by a sum on a fractional ideal a of F . Introduction
Put e ( z ) = e πiz for z ∈ C . It is known that the modular forms of SL ( Z )of weight 1/2 and 3/2 are the Dedekind eta function η ( z ) and its cubic power η ( z ) up to constant, respectively. Here, η ( z ) is given by η ( z ) = e ( z/ Y m ≥ (1 − e ( mz )) ( z ∈ h ) , where h is the upper half plane. It is known that η ( z ) = 12 X m ∈ Z χ ( m ) e ( mz/ , η ( z ) = 12 X m ∈ Z m χ ( m ) e ( mz/ . Here, χ and χ are the primitive character mod 12 and mod 4, respectively.Note that η ( z ) and η ( z ) are theta functions defined by a sum on Z .The function η ( z ) has the transformation formula with respect to modulartransformations (see [18, 19, 28]). Let (cid:16) ·· (cid:17) be the Jacobi symbol. We define (cid:16) ·· (cid:17) ∗ and (cid:16) ·· (cid:17) ∗ by (cid:16) cd (cid:17) ∗ = (cid:18) c | d | (cid:19) , (cid:16) cd (cid:17) ∗ = t ( c, d ) (cid:16) cd (cid:17) ∗ , t ( c, d ) = ( − c, d <
01 otherwise,for c ∈ Z \{ } and d ∈ Z + 1 such that ( c, d ) = 1. We understand (cid:18) ± (cid:19) ∗ = (cid:18) (cid:19) ∗ = 1 , (cid:18) − (cid:19) ∗ = − § For g ∈ SL ( R ) and z ∈ h , put(1) J ( g, z ) = √ d if c = 0 , d > −√ d if c = 0 , d < cz + d ) / if c = 0 , g = (cid:18) a bc d (cid:19) . Here, we choose arg( cz + d ) such that − π < arg( cz + d ) ≤ π . Then we have(2) η ( γ ( z )) = v η ( γ ) J ( γ, z ) η ( z ) , γ ( z ) = az + bcz + d ∈ h for any γ = (cid:18) a bc d (cid:19) ∈ SL ( Z ), where the multiplier system v η ( γ ) is givenby(3) v η ( γ ) = (cid:18) dc (cid:19) ∗ e (cid:18) ( a + d ) c − bd ( c − − c (cid:19) c : odd (cid:16) cd (cid:17) ∗ e (cid:18) ( a + d ) c − bd ( c −
1) + 3 d − − cd (cid:19) c : even.It is natural to ask the following problem. When does a Hilbert modulartheta series of weight 1/2 with respect to SL ( o ) exist? Here, o is thering of integers of a totally real number field F . In 1983, Feng [4] studiedthis problem. She gave a sufficient condition for the existence of a Hilbertmodular theta series of weight 1/2 with respect to SL ( o ) and constructedcertain Hilbert modular theta series. These series are defined by a sum on o . Let K be a real quadratic field and d K the discriminant of K . Gundlach[6, p.30], [7, Remark 4.1.] showed that if d K ≡ Q ( √ D ), D ≡ a of F . Let v be a placeof F and F v the completion of F at v . When v is a finite place, we write v < ∞ . When v is an infinite place, we have F v ≃ R and write v | ∞ . Let A be the adele ring of F .Let n = [ F : Q ] and ι v : F → F v be the embedding for any v . The entry-wise embeddings of SL ( F ) into SL ( F v ) are also denoted by ι v . The meta-plectic group of SL ( F v ) is denoted by ^ SL ( F v ), which is a nontrivial doublecovering group of SL ( F v ). Set-theoretically, it is { [ g, τ ] | g ∈ SL ( F v ) , τ ∈{± }} . Its multiplication law is given by [ g, τ ][ h, σ ] = [ gh, τ σc ( g, h )] for[ g, τ ] , [ h, σ ] ∈ ^ SL ( F v ), where c ( g, h ) is the Kubota 2-cocycle on SL ( F v ).Put [ g ] = [ g, {∞ , · · · , ∞ n } be the set of infinite places of F . Put ι i = ι ∞ i for1 ≤ i ≤ n . We embed SL ( F ) into SL ( R ) n by r ( ι ( r ) , · · · , ι n ( r )). Wedenote the embedding of SL ( F ) into SL ( A ) by ι . Let A f be the finite N MULTIPLIER SYSTEMS AND THETA FUNCTIONS 3 part of A and ι f : SL ( F ) → SL ( A f ) the projection of the finite part. Theembedding of F into A f is also denoted by ι f .Let ^ SL ( A ) be the adelic metaplectic group, which is a double covering ofSL ( A ). Let ˜ H be the inverse image of a subgroup H of SL ( A ) in ^ SL ( A ).It is known that SL ( F ) can be canonically embedded into ^ SL ( A ). Theembedding ˜ ι is given by g ([ ι v ( g )]) v for each g ∈ SL ( F ). We define themaps ˜ ι f : SL ( F ) → ^ SL ( A f ) and ˜ ι ∞ : SL ( F ) → ^ SL ( F ∞ ) by˜ ι f ( g ) = ([ ι v ( g )]) v< ∞ × ([1 ]) v |∞ , ˜ ι ∞ ( g ) = ([1 ]) v< ∞ × ([ ι i ( g )]) v |∞ , where 1 is the unit matrix of size 2. Then we have ˜ ι ( g ) = ˜ ι f ( g )˜ ι ∞ ( g ) forany g ∈ SL ( F ).Let Γ ⊂ SL ( o ) be a congruence subgroup. A map v : Γ → C × is saidto be a multiplier system of half-integral weight if v ( γ ) Q ni =1 J ( ι i ( γ ) , z i ) isan automorphy factor for Γ × h n , where J is the function in (1). We obtainan equivalent condition that v is a multiplier system of half-integral weight.Let K Γ be the closure of ι f (Γ) in SL ( A f ) and ˜ K Γ the inverse image of K Γ in ^ SL ( A f ). Let λ : ˜ K Γ → C × be a genuine character. Put v λ ( γ ) = λ (˜ ι f ( γ ))for γ ∈ Γ. Then v λ is a multiplier system of half-integral weight for Γ.Now suppose that v : Γ → C × is a multiplier system of half-integralweight. We obtain an equivalent condition that there exists a genuine char-acter λ : ˜ K Γ → C × such that v λ = v . Put K f = Q v< ∞ SL ( o v ), which o v isthe ring of integers of F v . Proposition 2. If F = Q , then any multiplier system v of half-integralweight of any congruence subgroup Γ ⊂ SL ( o ) is obtained from a genuinecharacter of ˜ K Γ . Proposition 3.
Let v be a multiplier system of half-integral weight forSL ( o ). Then there exists a genuine character λ : ˜ K f → C × such that v λ = v . Corollary 2.
There exists a multiplier system v of half-integral weight forSL ( o ) if and only if 2 splits completely in F/ Q . There exists a genuinecharacter of ^ SL ( o v ) for any v < ∞ , provided that this condition holds.Let ψ : A /F → C × be an additive character such that its v -component ψ v ( x ) equals e ( x ) for any v | ∞ . Put ψ β ( x ) = ψ ( βx ) and ψ β,v ( x ) = ψ v ( βx )for β ∈ F × . The Schwartz space of F v is denoted by S ( F v ). Let ω ψ β ,v bethe Weil representation of the metaplectic group ^ SL ( F v ) on S ( F v ) corre-sponding to ψ β,v .In the case v < ∞ , we shall determine the genuine characters of themetaplectic group ^ SL ( o v ). Let λ v be a genuine character of ^ SL ( o v ). Thespace ( ω ψ β ,v , S ( F v )) λ v is defined by a set of f ∈ S ( F v ) such that ω ψ β ,v ( g ) f = λ v ( g ) − f for any g ∈ ^ SL ( o v ). We determine the space completely.In the case v | ∞ , let λ v be a genuine character of the metaplecticgroup ^ SO(2), where SO(2) is a set of (cid:18) a b − b a (cid:19) ∈ SL ( R ). The space HIROSHI NOGUCHI ( ω ψ β ,v , S ( R )) λ v is defined by a set of f ∈ S ( R ) such that ω ψ β ,v ( g ) f = λ v ( g ) f for any g ∈ ^ SO(2). We have an irreducible decomposition ω ψ β ,v = ω + ψ β ,v ⊕ ω − ψ β ,v , where ω + ψ β ,v (resp. ω − ψ β ,v ) is an irreducible representation of the set of even(resp. odd) functions in S ( R ) (see [14, Lemma 2.4.4]).If β <
0, there exist no lowest weight vectors of ω + ψ β ,v or ω − ψ β ,v . If β > e ( iι v ( β ) x ) (resp. xe ( iι v ( β ) x )) is the lowest weight vector of ω + ψ β ,v (resp. ω − ψ β ,v ) of weight 1/2 (resp. 3/2) (see [14, Lemma 2.4.4]).Let λ ∞ , / be a genuine character of lowest weight 1/2 with respect to( ω + ψ β ,v , S ( R )) and λ ∞ , / of lowest weight 3/2 with respect to ( ω − ψ β ,v , S ( R )).Now suppose that 2 splits completely in F/ Q . The set of totally positiveelements of F is denoted by F × + . Assume that β ∈ F × + in order that thereexists a lowest weight vector of ( ω + ψ β ,v , S ( R )) or ( ω − ψ β ,v , S ( R )) for any v < ∞ .We fix ω ψ β ,v and λ v for any v . Here, we assume that λ v = λ ∞ , / or λ v = λ ∞ , / for any v | ∞ . Put K = K f × Q v |∞ SO(2). Let λ : ˜ K → C × be agenuine character such that its v -component is λ v . Let S ( A ) be the Schwartzspace of A . The space ( ω ψ β , S ( A )) λ is defined by a set of φ = Q v φ v ∈ S ( A )such that φ v ∈ ( ω ψ β ,v , S ( F v )) λ v for any v . We determine when there existsa nonzero φ ∈ ( ω ψ β , S ( A )) λ . We define the theta function Θ φ byΘ φ ( g ) = X ξ ∈ F ω ψ β ( g ) φ ( ξ )for φ ∈ S ( A ) and g ∈ ^ SL ( A ), where ω ψ β ( g ) φ ( ξ ) = Q v ω ψ β ,v ( g v ) φ v ( ξ ). Theproduct is essentially a finite product. If φ ∈ ( ω ψ β , S ( A )) λ for λ such thatthe lowest weight of ( ω ψ β , S ( A )) λ is 1/2 (resp. 3/2), then it is known thatΘ φ is a Hilbert modular form of weight 1/2 (resp. 3/2).For 1 ≤ i ≤ n , put λ ∞ i = λ ∞ ,w i , where w i = 1 / /
2. Put S ∞ = {∞ i | w i = 3 / } and S = { v < ∞ | F v = Q } . Let p v be the maximal ideal of o v and q v the order of o v / p v . Put T = { v < ∞ | q v = 3 } . We denote the orderof a set S by | S | . Let G be the set of triplets ( β, S , a ) of β ∈ F × + , a subset S ⊂ T and a fractional ideal a of F satisfying the conditions | S | + | S | + | S ∞ | ∈ Z and (8 β ) d Y v ∈ S p v = a , where d is the different of F/ Q . We define an equivalence relation ∼ on G by( β, S , a ) ∼ ( β ′ , S ′ , a ′ ) ⇐⇒ S = S ′ , β ′ = γ β, a ′ = γ a for some γ ∈ F × . We determine when there exists a nonzero Θ φ . Recall that if q v is odd,the double covering ^ SL ( F v ) → SL ( F v ) splits on SL ( o v ). We denote theimage of g ∈ SL ( o v ) under the splitting by [ g, s ( g )]. Thus if q v is odd, there N MULTIPLIER SYSTEMS AND THETA FUNCTIONS 5 exists a genuine character ǫ v : ^ SL ( o v ) → C × satisfying ǫ v ([ g, s ( g )]) = 1 forany g ∈ SL ( F v ). Now we set S = { v | q v = 3 , λ v = ǫ v } . Theorem 1.
Suppose that 2 splits completely in F/ Q . Let β ∈ F × + , λ : ˜ K → C × and w , . . . , w n ∈ { / , / } be as above. Then there exists φ = Q v φ v ∈ ( ω ψ β , S ( A )) λ such that Θ φ = 0 if and only if there exists afractional ideal a of F such that ( β, S , a ) ∈ G .Put H = Y v ∈ T p e v v | X v ∈ T e v ∈ Z . Let Cl + be the narrow ideal class group of F . Put Cl +2 = { c | c ∈ Cl + } .We denote the image of the group H (resp. b ∈ Cl + ) in Cl + / Cl +2 by ¯ H (resp. [ b ]). Theorem 2.
Suppose that 2 splits completely in F/ Q . Let w , . . . , w n ∈{ / , / } be as above.(1) Suppose that | S | + | S ∞ | is even. Then there exists ( β, S , a ) ∈ G ifand only if [ d ] ∈ ¯ H .(2) Suppose that | S | + | S ∞ | is odd. Then there exists ( β, S , a ) ∈ G ifand only if T = ∅ and [ dp v ] ∈ ¯ H . Here, v is any fixed element of T .Now suppose that there exists ( β, S , a ) ∈ G . Replacing β with βγ and a with ( a γ ) in (16), respectively, we may assume ord v a = 0 for v ∈ S ∪ S .For v ∈ S ∪ S , put f v ( x ) = x ∈ p v − x ∈ − p v f = Y v ∈ S ∪ S f v × Y v< ∞ ,v / ∈ S ∪ S ch a − v , where a v = ao v . Put φ = f × Q ni =1 f ∞ ,i , where f ∞ ,i ( x ) = x w i − (1 / e ( iι i ( β ) x )for x ∈ R and w i ∈ { / , / } . By Theorem 1, there exists Θ φ = 0 of weight w = ( w , · · · , w n ). Theorem 3.
Let φ and Θ φ be as above. We define a theta function θ φ : h n → C by θ φ ( z ) = X ξ ∈ a − f ( ι f ( ξ )) Y ∞ i ∈ S ∞ ι i ( ξ ) n Y i =1 e ( z i ι i ( βξ )) . for z = ( z , · · · , z n ) ∈ h . Then θ φ is a nonzero Hilbert modular form ofweight w for SL ( o ) with respect to a multiplier system. Every theta functionof weight w for SL ( o ) with a multiplier system may be obtained in this way.In particular, when F = Q , we obtain η ( z ) and η ( z ) as θ φ ( z ) up toconstant. HIROSHI NOGUCHI
This paper is organized as follows. In Section 2, we determine the numberof the genuine characters of the metaplectic group ^ SL ( o ), where o is thering of integers of a finite extension F of Q p . Moreover, we determine thedimension of a space ( ω ψ β , S ( F )) λ for a genuine character λ of ^ SL ( o ) andthe Weil representation ω ψ β of ^ SL ( F ) on S ( F ). In Section 3, we studythe multiplier systems of half-integral weight of a congruence subgroup ofSL ( o ), where o is the ring of integers of a totally real number field F .In Section 4, we define theta functions Θ φ of ^ SL ( A ) and prove our maintheorems. Moreover, we obtain theta functions θ φ ( z ) of h n and determinethe number of the equivalence classes of the set G . In Section 5, we givesome examples in the case F = Q or F is a real quadratic field. Acknowledgment.
The author thanks his supervisor Tamotsu Ikeda forsuggesting the problem and for his helpful advice, and thanks Masao Oi andShuji Horinaga for their sincere and useful comments.2.
Local theory, genuine characters of ^ SL ( o )Let F be a finite extension of Q p until the end of this section. Let o bethe ring of integers of F and p the maximal ideal of o . Let q be the order ofthe residue field o / p and d the different of F/ Q p .For g = (cid:18) a bc d (cid:19) ∈ SL ( F ), put x ( g ) = c if c = 0 and x ( g ) = d if c = 0.The Kubota 2-cocycle on SL ( F ) is defined by c ( g, h ) = h x ( g ) x ( gh ) , x ( h ) x ( gh ) i F for g, h ∈ SL ( F ), where h· , ·i F is the quadratic Hilbert symbol for F . Let ^ SL ( F ) be the metaplectic group of SL ( F ). Set-theoretically, it is { [ g, τ ] | g ∈ SL ( F ) , τ ∈ {± }} . Its multiplication law is given by [ g, τ ][ h, σ ] = [ gh, τ σc ( g, h )]. This is anontrivial double covering group of SL ( F ). Put [ g ] = [ g, H of SL ( F ), the inverse image of H in ^ SL ( F ) is denoted by ˜ H . A function ǫ F : ^ SL ( o ) → C is genuine if ǫ F ([1 , − γ ) = − ǫ F ( γ ) for any γ ∈ ^ SL ( o ).We determine the number of the genuine characters of ^ SL ( o ). For a ∈ F × , τ = ± b, c ∈ F , put m ( a, τ ) = (cid:20)(cid:18) a a − (cid:19) , τ (cid:21) , u + ( b ) = (cid:20)(cid:18) b (cid:19)(cid:21) ,u − ( c ) = (cid:20)(cid:18) c (cid:19)(cid:21) , N = (cid:20)(cid:18) −
11 0 (cid:19)(cid:21) . For k ∈ Z such that k ≥
0, we define the subgroups U + ( p k ), U − ( p k ) and˜ A of ^ SL ( o ) by U + ( p k ) = { u + ( b ) | b ∈ p k } , U − ( p k ) = { u − ( c ) | c ∈ p k } and ˜ A = { m ( a, τ ) | a ∈ o × } , respectively. Note that ^ SL ( o ) is generated by U + ( o ), N and m (1 , − N MULTIPLIER SYSTEMS AND THETA FUNCTIONS 7
Lemma 1.
Put M = min { ord( a − | a ∈ o × } . Then we have M = q ≥ q = 3)2 ( q = 2 , F = Q )3 ( F = Q ) . Proof.
Let π be a prime element of F . If q ≥
4, then there exists a ∈ o × such that a − ∈ o × . Thus we have M = 0. If q = 3, then a − ∈ p forany a ∈ o × . Since ( π + 1) − π ( π + 2) / ∈ p , we have M = 1.In the case q = 2, a − a − a + 1) ∈ p for any a ∈ o × . If F = Q ,then we have 2 ∈ p . Since ( π + 1) − π ( π + 2) / ∈ p , we have M = 2.It is well-known that M = 3 if F = Q . (cid:3) The derived group of a group G is denoted by D ( G ). Since [ m ( a, τ ) , u + ( b )] = u + (( a − b ) and [ m ( a, τ ) , u − ( c )] = u − (( a − − c ) hold, we have(4) U + ( p M ) , U − ( p M ) ⊂ D ( ^ SL ( o ))by Lemma 1. Lemma 2.
Suppose that q is even. Then there exists r ∈ o such that h r, x i F = ( − ord x for x ∈ F × . Proof.
Put r = 1 + 4 c for c ∈ o . We show that there exists c such that F ( √ r ) /F is an unramified quadratic extension. We denote the residue fieldof a local field L by k ( L ) and the image of an element u of the ring of integersof F ( √ r ) in k ( F ( √ r )) by ¯ u .We define a map p : k ( F ) → k ( F ) by p ( t ) = t − t for t ∈ k ( F ). Wehave p ( t ) = p (1 − t ) = p ( s ) for any s ∈ k ( F ) \{ t, − t } . Since [ k ( F ) : p ( k ( F ))] = 2, there exists c such that ¯ c / ∈ p ( k ( F )). Then it is known thata polynomial X − X − ¯ c is irreducible over k ( F ). Put y = (1 − √ r ) / f ( X ) = X − X − c ∈ o [ X ]. Since f ( y ) = 0 and f ′ ( y ) = 2 y − −√ r =0, k ( F )(¯ y ) /k ( F ) is a quadratic extension and k ( F ( √ r )) = k ( F ( y )) equals k ( F )(¯ y ). Therefore F ( √ r ) /F is an unramified quadratic extension. (cid:3) Lemma 3.
Suppose that q is even and F = Q . Then there exist no genuinecharacters of ^ SL ( o ). Proof.
Let b, c ∈ o such that r = 1 − bc ∈ o × and ζ = h r, b i F . We have(5)[ u − ( c ) , u + ( b )] = (cid:20)(cid:18) r b c − bc bc + b c (cid:19)(cid:21) = u − ( − r − bc ) m ( r, ζ ) u + ( r − b c ) . When F/ Q is a ramified extension, we assume that b, c ∈ o such that r satisfies the condition in Lemma 2. We have U + ( p ) , U − ( p ) ⊂ D ( ^ SL ( o ))by (4). Then we have m ( r, ζ ) ∈ D ( ^ SL ( o )) by (5). Let π be a prime elementof F . Set b ′ = bπ and c ′ = cπ − , which lie in p . We have h − b ′ c ′ , b ′ i F = h r, bπ i F = − ζ . Thus we have m (1 , − ∈ D ( ^ SL ( o )) and there exist nogenuine characters of ^ SL ( o ). HIROSHI NOGUCHI
Assume that F/ Q is an unramified extension and that F = Q . Wehave U + ( o ), U − ( o ) ⊂ D ( ^ SL ( o )) by (4). Substituting 1 for c in (5), wehave m (1 − b, ζ ) ∈ D ( ^ SL ( o )), whenever 1 − b ∈ o × . Since ζ = h − b, b i F equals 1, we have m (1 − b, ∈ D ( ^ SL ( o )). Similarly, substituting − c and replacing b with − b in Equation (5), we have m (1 − b, h − b, − i F ) ∈ D ( ^ SL ( o )). Thus it suffices to show that there exists b such that h − b, − i F equals − F/ Q is unramified, F ( √− /F is a ramified extension. Thus thereexists u ∈ o × such that h u, − i F = −
1. Since [ o × : 1 + p ] = q − u ∈ p . Then there exists b ∈ p such that u = 1 − b satisfies h u, − i F = − (cid:3) An additive character e p of Q p is defined by e p ( x ) = e ( − x ) for any x ∈ Z [1 /p ]. We define a nontrivial additive character ψ β of F by x e p (Tr F/ Q p ( βx )) for β ∈ F × . The order of ψ β is denoted by ord ψ β ∈ Z ,which is defined by ψ β ( p − ord ψ β ) = 1 and ψ β ( p − ord ψ β − ) = 1. We haveord ψ β = ord d + ord β .Let S ( F ) be the Schwartz space of F . The Fourier transformation ˆ φ of φ ∈ S ( F ) is defined by ˆ φ ( x ) = R F φ ( y ) ψ β ( xy ) dy . Here, dy is self-dual on theFourier transformation. In other words, dy is the Haar measure such thatthe Plancherel’s formula R F | φ ( y ) | dy = R F | ˆ φ ( y ) | dy holds, where | · | is theabsolute value on F .We denote the characteristic function of a subset A of a set X by ch A . Inthe case F = Q p , we have the volume vol( p m Z p ) of p m Z p equals p − m − (ord β/ and \ ch p m Z p = vol( p m Z p ) ch p − ( m +ord β ) Z p for m ∈ Z .Put A φ = R F φ ( x ) ψ β ( ax ) dx and B φ = R F ˆ φ ( x ) ψ β ( − x / a ) dx for a ∈ F × and φ ∈ S ( F ). There exists a constant α ψ β ( a ) ∈ C called the Weil constantsuch that A φ = α ψ β ( a ) | a | − / B φ holds. It is known that α ψ β ( ab ) = α ψ β ( a )for a, b ∈ F × and that α ψ β ( a ) = α ψ ( aβ ), where ψ = ψ . Moreover, we have α ψ β ( − a ) = α ψ β ( a ) and α ψ β ( a ) = 1.The Weil representation ω ψ β is a representation of ^ SL ( F ) on S ( F ). For φ ∈ S ( F ), we have(6) ω ψ β ( m ( a, τ )) φ ( x ) = τ α ψ β (1) α ψ β ( − a ) | a | / φ ( ax ) ω ψ β ( u + ( b )) φ ( x ) = ψ β ( bx ) φ ( x ) ω ψ β ( N ) φ ( x ) = | | / α ψ β ( −
1) ˆ φ ( − x ) . Since ^ SL ( F ) is generated by the above elements, ω ψ β is determined by theseformulas. In particular, we have ω ψ β ([ g, τ ]) φ = τ ω ψ β ([ g ]) φ .We define a map s : SL ( o ) → {± } by(7) s ( g ) = c ∈ o × h c, d i F c ∈ p \{ }h− , d i F c = 0 for g = (cid:18) a bc d (cid:19) ∈ SL ( o ) . N MULTIPLIER SYSTEMS AND THETA FUNCTIONS 9 If q is odd, we have s ( g ) = ( cd = 0 h c, d i ord cF cd = 0 . Recall that the double covering ^ SL ( F ) → SL ( F ) splits on SL ( o ) if andonly if q is odd. The splitting is given by g [ g, s ( g )]. Thus if q is odd, amap ǫ F : ^ SL ( o ) → C × defined by ǫ F ([ g, τ ]) = τ s ( g ) is a genuine character.If ord ψ β = 0, we have ω ψ β ([ g, τ ]) ch o = ǫ F ([ g, τ ]) − ch o . Lemma 4.
Suppose that q is odd. Then there exists a genuine character ǫ F : ^ SL ( o ) → C × such that ω ψ β ([ g, τ ])ch o = ǫ F ([ g, τ ]) − ch o if ord ψ β = 0.If q ≥
5, then it is a unique genuine character of ^ SL ( o ). Proof.
We already proved the first part of this lemma. If q ≥
5, then by(4) we have U + ( o ), U − ( o ) ⊂ D ( ^ SL ( o )). Since N = u + ( − u − (1) u + ( − ( o ) ab is trivial. Thus there exists a unique genuine character ǫ F of ^ SL ( o ). (cid:3) For a ∈ Z , we define a subset S ( p a ) of S ( F ) by S ( p a ) = { f ∈ S ( F ) | Supp f ⊂ p a } . For a ≤ b ∈ Z , put S ( p a / p b ) = { f ∈ S ( p a ) | f ( x + t ) = f ( x )for any t ∈ p b } . For f ∈ S ( F ) \{ } , there exists a pair ( a, b ) such that f ∈ S ( p a / p b ). Lemma 5.
Suppose that q = 3 (resp. F = Q ). If ord ψ β = − − ^ SL ( o ) preserves S ( o / p ) with respect to ω ψ β . We define f ∈ S ( o / p ) by(8) f ( x ) = x ∈ p − x ∈ − p . Then the subspace of odd functions is C f ⊂ S ( o / p ) and there exists agenuine character µ β of ^ SL ( o ) such that ω ψ β ([ g, τ ]) f = µ β ([ g, τ ]) − f .In the case q = 3, there exist three genuine characters of ^ SL ( o ), ǫ F and µ β , where µ β extends over all elements β such that ord ψ β = −
1. More-over, the value µ β ( u + (1)) = ψ β ( −
1) is a primitive 3rd root of unity, whichdetermines µ β . Proof.
Suppose that F = Q and that ord ψ β = ord β = −
3. It is clear that ^ SL ( o ) preserves S ( o / p ) with respect to ω ψ β . If φ ∈ S ( o / p ) is an oddfunction, then φ satisfies φ ( x ) = φ ( − x ) = − φ ( x ) for any x ∈ p . Thus wehave φ ( p ) = 0. Since F = Q , we have o × = (1 + 2 p ) ∪ ( − p ) and then φ ( x ) = φ (1) f ∈ C f .Thus there exists a genuine character µ β of ^ SL ( o ) such that ω ψ β ([ g, τ ]) f = µ β ([ g, τ ]) − f . Since u − ( −
1) =
N u + (1) N − and N = u + ( − u − (1) u + ( − µ β ( u + (1)) determines µ β . Suppose that q = 3 and that ord ψ β = −
1. Then we prove the first partof the lemma similar to the case above. By [10, § ( o ) ab has order3. Thus there exist three genuine characters of ^ SL ( o ). We have f ( x ) = 0if and only if x ∈ o × . By (6), we have ω ψ β ( u + ( b )) f ( x ) = ψ β ( bx ) f ( x ) for b ∈ o . If f ( x ) = 0, since we have x − ∈ p by Lemma 1, µ β ( u + ( b )) − = ψ β ( bx ) = ψ β ( b ). In particular, µ β ( u + (1)) = ψ β ( −
1) is a primitive 3rd rootof unity.For γ ∈ F such that ord ψ γ = −
1, if µ β = µ γ , then we have ψ β (1) = ψ γ (1)and then β/γ ∈ p . Since we have [ o × : (1+ p )] = 2, the genuine charactersof ^ SL ( o ) are ǫ F and µ β , where µ β extends over all elements β such thatord ψ β = − (cid:3) Lemma 6.
In Lemma 5, suppose that F = Q and that ord β = −
3. Thenthe value µ β ( u + (1)) = e ( β ) is a primitive 8th root of unity, which determines µ β . Moreover, there exist four genuine characters µ β of ^ SL ( Z ), where µ β extends over all elements β such that ord β = − Proof.
By Lemma 5, there exists a genuine character of ^ SL ( Z ). SinceSL ( Z ) ab has order 4 by [10, § ^ SL ( Z ) is 4. We have ω ψ β ( u + ( b )) f ( x ) = e ( − βbx ) f ( t ) for f in (8) and b ∈ Z by (6). If f ( x ) = 0, then we have x − ∈ Z and e ( − βbx ) = e ( − βb ).In particular, µ β ( u + (1)) = e ( β ) is a primitive 8th root of unity.For γ ∈ Q such that ord γ = −
3, if µ β = µ γ , then we have e ( β ) = e ( γ )and then β/γ ∈ Z . Since we have [ Z × : (1 + 8 Z )] = 4, there existfour genuine characters µ β of ^ SL ( Z ), where µ β extends over all elements β such that ord β = − (cid:3) Let µ β be a nontrivial genuine character in Lemma 5 or Lemma 6. Thenwe have µ β ([ g ]) = s ( g ) κ ( β, g ) g = (cid:18) a bc d (cid:19) ∈ SL ( o ) , where κ ( β, g ) is a continuous function for g . In the case F = Q andord β = −
3, we have(9) κ ( β, g ) = ( ψ β ( − ( a + d ) c + 3 c ) c ∈ Z × ψ β (( c − b ) d − d − c ∈ Z . In the case q = 3 and ord ψ β = −
1, we have(10) κ ( β, g ) = ψ β ( − ( a + d ) c + bd ( c − . Remark.
Suppose that K ′ is a compact open subgroup of SL ( o ). Let λ ′ : ˜ K ′ → C × be a genuine character. Then one can show that there existsa continuous function κ ′ on K ′ such that λ ′ ([ g ]) = s ( g ) κ ′ ( g ) for any g ∈ K ′ .As we do not need this result for the rest of this paper, we omit a proof.Put K = SL ( o ) and G =SL ( F ). It is known that K (resp. ˜ K ) is acompact open subgroup of G (resp. ˜ G ). Let ( π, V ) be an irreducible smooth N MULTIPLIER SYSTEMS AND THETA FUNCTIONS 11 representation of ˜ G . For a character λ of ˜ K , we define a set ( π, V ) λ by( π, V ) λ = { f ∈ V | π ( g ) f = λ ( g ) − f for any g ∈ ˜ K } . In particular, we consider ( ω ψ β , S ( F )) λ for a genuine character λ : ˜ K → C × such that λ ( u + (1)) = 1. Since λ ( u + (1)) = 1, λ is the character µ β inLemma 5 or Lemma 6. In particular, we have q = 3 or F = Q . When q = 3(resp. F = Q ), we assume that ord ψ β = − − Proposition 1.
The representation c -Ind ˜ G ˜ K λ is irreducible supercuspidal.We have dim C ( π, V ) λ = ( π = c -Ind ˜ G ˜ K λ Proof.
Put λ g ( x ) = λ ( gxg − ) for g ∈ ˜ G . We shall prove that(11) Hom g − Kg ∩ K ( λ g , λ ) = 0 for g / ∈ ˜ K. Put m ( a ) = diag( a, a − ) for a ∈ F × . Since it is known thatSL ( F ) = ∞ X n =0 Km ( π n ) K, we have only to consider the case g = m ( π n ) for n >
0. Since u + (1) ∈ g − Kg ∩ K , we have λ ( u + (1)) = λ g ( u + (1)), which proves (11).It is known that (11) implies the first assertion (see [2, § case but the proof is valid in our case. By [2, § ˜ G ( c -Ind ˜ G ˜ K λ, π ) ≃ Hom ˜ K ( λ, π ), which completes the proof. (cid:3) It is known that ω ψ β = ω + ψ β ⊕ ω − ψ β , where ω + ψ β (resp. ω − ψ β ) is the restriction of ω ψ β to the even (resp. odd)functions. Note that these restrictions are irreducible but not isomorphic.By Proposition 1, we have ω − ψ β ≃ c-Ind ˜ G ˜ K µ β . Since λ ( u + (1)) = 1, we havedim( ω + ψ β , S ( F )) λ = 0 by Proposition 1. Since we assume that ord ψ β = − q = 3 and that ord β = − F = Q , we have(12) dim C ( ω ψ β , S ( F )) λ = ( λ = µ β q is odd. By Lemma 4, there exists a genuinecharacter ǫ F : ^ SL ( o ) → C × and we have ω ψ β ([ g, τ ]) ch o = ǫ F ([ g, τ ]) − ch o ,where ord ψ β = 0. Lemma 7.
Put T β = ( ω ψ β , S ( F )) ǫ F . Then we have T β = ( C ch p − ord ψ β / if ord ψ β ≡ Proof.
Put D = ord ψ β . Suppose that D = 0. Then we have ǫ F ( U + ( o )) =1 and it is clear that ^ SL ( o ) preserves S ( o / o ) = C ch o with respect to ω ψ β . Then we have T β = C ch o = C ch p − D/ . Since we have ω ψ βt ( g ) = ω ψ β ( m ( t, g m ( t, − ) for t ∈ F × and g ∈ ^ SL ( o ), the same is true for T βt for any t ∈ F × .Assume that φ ∈ T β \{ } with D = 1. Then we have ω ψ β ( h ) φ = ǫ F ( h ) − φ .By (6), we have ψ β ( bx ) = 1 for any b ∈ o × when φ ( x ) = 0. In particular,since ord bx ≥ ord x ≥ −
1, we have φ ∈ S ( o ).We assume φ ∈ S ( p a / p b ) such that a ≥ b is minimal. Acalculation of the Fourier transformation shows that ˆ φ ∈ S ( p − − b / p − − a ).Since α ψ β ( −
1) ˆ φ ( − x ) = φ ( x ) by (6), we have ˆ φ ∈ S ( p a / p b ). Then a = − − b is less than 0, which contradicts a ≥ (cid:3) Set F × = { x | x ∈ F × } . Assume that q = 3 or F = Q . By Lemma 5and Lemma 6, there exist genuine characters µ β of ^ SL ( o ). Lemma 8.
When q = 3 (resp. F = Q ), we put T β = ( ω ψ β , S ( F )) µ γ , where γ ∈ F × such that ord ψ γ = − − T β = ( β/γ ∈ F × β/γ ∈ p (resp. 1 + 8 Z ), we have T β = C f , where f is the function in (8). Proof.
We prove the lemma in the case q = 3. The proof for F = Q issimilar. Put D = ord ψ β . Then we may assume that D ∈ { , − } in thesame way as the proof of Lemma 7. By Proposition 1 and (12), we havedim T β = 0 when D = 0. Suppose that D = − φ ∈ T β is nonzero.Then by Lemma 5, we have φ ∈ C f . Lemma 5 shows that f lies in T β if andonly if β/γ ∈ p . We have 1 + p ⊂ F × , which completes the proof. (cid:3) Multiplier systems for SL ( o )From now on, let F be a totally real number field such that [ F : Q ] = n .Let v be a place of F and A the adele ring of F . We denote the completionof F at v by F v . If v is an infinite place, we write v | ∞ . Otherwise, wewrite v < ∞ . For v < ∞ , let o v , p v and d v be the ring of integers of F v , themaximal ideal of o v and the different of F v / Q p , respectively.For any v , let ι v : F → F v be the embedding. The entrywise embeddingsof SL ( F ) into SL ( F v ) are also denoted by ι v . Let {∞ , · · · , ∞ n } be theset of infinite places of F . Put ι i = ι ∞ i for 1 ≤ i ≤ n . We embed SL ( F )into SL ( R ) n by r ( ι ( r ) , · · · , ι n ( r )).We define the metaplectic group ^ SL ( R ) of SL ( R ) similar to the case F v / Q p . Let S be a finite set of places of F , which contains all places above2 and ∞ . Set SL ( A ) S = Y v ∈ S SL ( F v ) × Y v / ∈ S SL ( o v ) . N MULTIPLIER SYSTEMS AND THETA FUNCTIONS 13
The double covering of SL ( A ) S defined by the 2-cocycle Q v ∈ S c v ( g ,v , g ,v )is denoted by ^ SL ( A ) S , where c v is the Kubota 2-cocycle for SL ( F v ).Let s v : SL ( o v ) → {± } is the map s in (7) for v < ∞ . For a finiteset S ′ of places of F such that S ⊂ S ′ , we can define an embedding ι S ′ S : ^ SL ( A ) S → ^ SL ( A ) S ′ by[( g v ) , ζ ] h ( g v ) , ζ Y v ∈ S ′ \ S s v ( g v ) i . For v < ∞ , a map s v : SL ( o v ) → ^ SL ( o v ) is given by s v ( γ ) = [ γ, s v ( γ )]for γ ∈ SL ( o v ). The adelic metaplectic group ^ SL ( A ) is the direct limitlim −→ ^ SL ( A ) S . It is a double covering of SL ( A ) and there exists a canonicalembedding ^ SL ( F v ) → ^ SL ( A ) for each v . Let Q ′ v ^ SL ( F v ) be the restricteddirect product with respect to s v (SL ( o v )). Then there is a canonical sur-jection Q ′ v ^ SL ( F v ) → ^ SL ( A ). The image of ( g v ) v ∈ Q ′ v ^ SL ( F v ) is alsodenoted by ( g v ) v . Note that for a given g ∈ ^ SL ( A ), the expression g = ( g v ) v is not unique.We denote the embedding of SL ( F ) into SL ( A ) by ι . The finite part ofSL ( A ) is denoted by SL ( A f ). Let ι f : SL ( F ) → SL ( A f ) be the projectionof the finite part and ι ∞ : SL ( F ) → SL ( F ∞ ) = SL ( R ) n that of the infinitepart. Then we have ι ( g ) = ι f ( g ) ι ∞ ( g ) for any g ∈ SL ( F ). The embeddingof F into A f is also denoted by ι f .It is known that SL ( F ) can be canonically embedded into ^ SL ( A ). Theembedding ˜ ι is given by g ([ ι v ( g )]) v for each g ∈ SL ( F ). We define themaps ˜ ι f : SL ( F ) → ^ SL ( A f ) and ˜ ι ∞ : SL ( F ) → ^ SL ( F ∞ ) by˜ ι f ( g ) = ([ ι v ( g )]) v< ∞ × ([1 ]) v |∞ , ˜ ι ∞ ( g ) = ([1 ]) v< ∞ × ([ ι i ( g )]) v |∞ . Then we have ˜ ι ( g ) = ˜ ι f ( g )˜ ι ∞ ( g ) for any g ∈ SL ( F ).For γ = [ g, τ ] ∈ ^ SL ( R ), g = (cid:18) a bc d (cid:19) and z ∈ h , ˜ j : ^ SL ( R ) × h → C is anautomorphy factor given by(14) ˜ j ( γ, z ) = τ √ d if c = 0 , d > , − τ √ d if c = 0 , d < ,τ ( cz + d ) / if c = 0 . Here, we choose arg( cz + d ) such that − π < arg( cz + d ) ≤ π . Note that˜ j ([ g, τ ] , z ) is the unique automorphy factor such that ˜ j ([ g, τ ] , z ) = j ( g, z ),where j ( g, z ) is the usual automorphy factor on SL ( R ) × h (see [9, § j ([ g ] , z ) = J ( g, z ), where J ( g, z ) is defined in (1). Definition 1.
Let Γ ⊂ SL ( o ) be a congruence subgroup. the map v = v ( γ ) : Γ → C × is said to be a multiplier system of half-integral weight if v ( γ ) Q ni =1 ˜ j ([ ι i ( γ )] , z i ) is an automorphy factor for Γ × h n , where ˜ j is theautomorphy factor in (14). We have ˜ j ( γ , γ ( z ))˜ j ( γ , z ) = ˜ j ( γ γ , z ) for γ , γ ∈ ^ SL ( R ). Replacing γ i with [ g i ] for i = 1 ,
2, we have(15) ˜ j ([ g ] , g ( z ))˜ j ([ g ] , z ) = c R ( g , g )˜ j ([ g g ] , z ) , where c R ( · , · ) is the Kubota 2-cocycle at infinite places. Lemma 9.
A function v : Γ → C × is a multiplier system of half-integralweight if and only if we have v ( γ ) v ( γ ) = c ∞ ( γ , γ ) v ( γ γ ) γ , γ ∈ Γ , where c ∞ ( γ , γ ) = Q ni =1 c R ( ι i ( γ ) , ι i ( γ )). Proof.
We have ι i ( γ ) ι i ( γ ) = ι i ( γ γ ) for any i . Thus (15) and Definition1 prove the lemma. (cid:3) Let K Γ ⊂ SL ( A f ) be the closure of ι f (Γ) in SL ( A f ). Then K Γ is acompact open subgroup and we have ι − f ( K Γ ) = Γ. Let ˜ K Γ be the inverseimage of K Γ in ^ SL ( A f ). Lemma 10.
Let λ : ˜ K Γ → C × be a genuine character. Put v λ ( γ ) = λ (˜ ι f ( γ ))for γ ∈ Γ. Then v λ is a multiplier system of half-integral weight for Γ. Proof.
For γ , γ ∈ Γ, we have ˜ ι ( γ )˜ ι ( γ ) = ˜ ι ( γ γ ). The left-hand sideequals ˜ ι f ( γ )˜ ι ∞ ( γ )˜ ι f ( γ )˜ ι ∞ ( γ ). Since ˜ ι ∞ ( g ) = ([ ι i ( g )]) i =1 , ··· ,n for g ∈ SL ( F ) and ˜ ι ∞ ( γ ) commutes with ˜ ι f ( γ ), we have˜ ι f ( γ )˜ ι f ( γ ) = ˜ ι f ( γ γ )[1 , c ∞ ( γ , γ )] . Since λ is genuine, Lemma 9 proves the lemma. (cid:3) For v < ∞ , the map s v is the splitting on K (4) v , where K (4) v = { γ = (cid:18) a bc d (cid:19) ∈ SL ( o v ) | c ≡ , d ≡ } . If K Γ ⊂ K (4) f = Q v< ∞ K (4) v , we may define a splitting s : K Γ → ^ SL ( A )by s ( γ ) = ( s v ( ι v ( γ ))) v< ∞ × ([1 ]) v |∞ . We consider it as a homomorphism. Then we have ˜ K Γ = s ( K Γ ) · { [1 , ± } .Note that s ( K Γ ) ⊂ ^ SL ( A f ) is a compact open subgroup.For any congruence subgroup Γ, a map v : Γ → C × is defined by v ( γ ) = Q v< ∞ s v ( ι v ( γ )), which is not always a multiplier system of half-integralweight for Γ. Corollary 1.
If Γ ⊂ Γ (4) = (cid:26)(cid:18) a bc d (cid:19) ∈ SL ( o ) | c ≡ , d ≡ (cid:27) ,then v is a multiplier system of half-integral weight for Γ. Proof.
Since Γ ⊂ Γ (4), we have K Γ ⊂ K (4) f . We define a genuine char-acter λ : ˜ K Γ → C × by λ ( s ( k )[1 , τ ]) = τ, k ∈ K Γ , τ ∈ {± } . N MULTIPLIER SYSTEMS AND THETA FUNCTIONS 15
Put v λ ( γ ) = λ (˜ ι f ( γ )) for γ ∈ Γ. Since s ( γ ) = ([ ι v ( γ ) , s v ( ι v ( γ ))]) v< ∞ , wehave v λ ( γ ) = λ ( s ( γ )[1 , v ( γ )]) = v ( γ ) . Therefore Lemma 10 proves the corollary. (cid:3)
Now suppose that Γ ⊂ SL ( o ) is a congruence subgroup and that v : Γ → C × is a multiplier system of half-integral weight. Lemma 11.
There exists a genuine character λ : ˜ K Γ → C × such that v λ = v if and only if there exists a congruence subgroup Γ ′ ⊂ Γ ∩ Γ (4) suchthat v ( γ ) = v ( γ ) for any γ ∈ Γ ′ . Proof.
Suppose that there exists a genuine character λ : ˜ K Γ → C × such that v λ = v . Since Ker λ and s (Γ (4)) are open in ^ SL ( A f ), the intersection isalso open. We denote its image in SL ( A f ) by K ′ . Then we have ˜ K ′ = s ( K ′ ) × { [1 , ± } . Put Γ ′ = ι − f ( K ′ ). Then we have v ( γ ) = v λ ( γ ) = v ( γ )for any γ ∈ Γ ′ .Conversely, suppose that there exists a congruence subgroup Γ ′ ⊂ Γ ∩ Γ (4) such that v ( γ ) = v ( γ ) for any γ ∈ Γ ′ . Then the closure K Γ ′ of ι f (Γ ′ )in SL ( A f ) is a compact open subgroup. Since ι f (Γ) is dense and K Γ ′ isopen in K Γ , we have K Γ = ι f (Γ) · K Γ ′ . For k ∈ ˜ K Γ , there exist γ ∈ Γ, k ′ ∈ K Γ ′ and τ ∈ {± } such that k = ˜ ι f ( γ ) s ( k ′ )[1 , τ ].We assume that k also equals ˜ ι f ( γ ) s ( k ′ )[1 , τ ] for γ ∈ Γ, k ′ ∈ K Γ ′ and τ ∈ {± } . Put ω = γ − γ . Then we have ω ∈ Γ ′ and˜ ι f ( γ ) = ˜ ι f ( γ )˜ ι f ( ω )[1 , c ∞ ( γ , ω )] , ˜ ι f ( ω ) = s ( ι f ( ω ))[1 , v ( ω )] . Then we have k = ˜ ι f ( γ ) s ( k ′ )[1 , τ ] = ˜ ι f ( γ ) s ( ι f ( ω ) k ′ )[1 , τ v ( ω ) c ∞ ( γ , ω )].Thus we have k ′ = ι f ( ω ) k ′ and τ = τ v ( ω ) c ∞ ( γ , ω ). Since v = v inΓ ′ and v ( γ ) = v ( γ ) v ( ω ) c ∞ ( γ , ω ) by Lemma 9, we have v ( γ ) τ = v ( γ ) τ .Then a function λ ( k ) = v ( γ ) τ is well-defined.Since λ ( k [1 , σ ]) = v ( γ ) τ σ = σλ ( k ) for σ ∈ {± } , λ is genuine. It sufficesto show that λ ( k k ) = λ ( k ) λ ( k ) for any k , k ∈ ˜ K Γ . There exist γ i ∈ Γ, k ′ i ∈ K Γ ′ and τ i ∈ {± } such that k i = ˜ ι f ( γ i ) s ( k ′ i )[1 , τ i ] for i = 1 ,
2. Thenwe have λ ( k ) λ ( k ) = v ( γ ) v ( γ ) τ τ . Replacing K Γ ′ with its sufficientlysmall subgroup, we may assume that s ( K Γ ′ ) is a normal subgroup of ˜ K Γ .Then we have˜ ι f ( γ ) − s ( k ′ )˜ ι f ( γ ) = s ( ι f ( γ ) − k ′ ι f ( γ )) ∈ s ( K Γ ′ ) . Since ˜ ι f ( γ )˜ ι f ( γ ) = ˜ ι f ( γ γ )[1 , c ∞ ( γ , γ )], λ ( k k ) equals v ( γ γ ) c ∞ ( γ , γ ) τ τ .By Lemma 9, we have λ ( k k ) = λ ( k ) λ ( k ), which proves the lemma. (cid:3) Proposition 2. If F = Q , then any multiplier system v of half-integralweight of any congruence subgroup Γ ⊂ SL ( o ) is obtained from a genuinecharacter of ˜ K Γ . Proof.
By Lemma 11, it suffices to show that there exists a congruencesubgroup Γ ′ ⊂ Γ ∩ Γ (4) such that v ( γ ) = v ( γ ) for any γ ∈ Γ ′ . Weassume that a congruence subgroup Γ satisfies Γ ⊂ Γ (4) by replacing Γwith Γ ∩ Γ (4). Since v ( γ ) / v ( γ ) is a character of Γ, we have v ( γ ) / v ( γ ) = 1 for any γ ∈ D (Γ). By the congruence subgroup property, D (Γ) contains acongruence subgroup Γ ′ (see [24, Corollary 3 of Theorem 2] or [10, § v ( γ ) = v ( γ ) for any γ ∈ Γ ′ , which proves this proposition. (cid:3) By Lemma 11 and Proposition 2, the multiplier system of half-integralweight of a congruence subgroup Γ associated to an automorphy factor inthe sense of Shimura [25] is obtained from a genuine character of ˜ K Γ . Lemma 12. If F = Q , then we have v ( g ) = (cid:18) dc (cid:19) ∗ c : odd (cid:16) cd (cid:17) ∗ c : even , g = (cid:18) a bc d (cid:19) ∈ SL ( Z ) . Proof.
In the case ( c, d ) = ( ± , (cid:18) c (cid:19) ∗ = v ( g ) = 1. In the case( c, d ) = (0 ,
1) (resp. (0 , − (cid:18) d (cid:19) ∗ = v ( g ) = 1(resp. − c = 0 and d ∈ Z + 1 satisfy ( c, d ) = 1, we have (cid:16) cd (cid:17) ∗ = (cid:18) c | d | (cid:19) , (cid:16) cd (cid:17) ∗ = t ( c, d ) (cid:18) c | d | (cid:19) , t ( c, d ) = ( − c, d <
01 otherwise.Suppose that cd = 0. Put u = c · − ord c . Then we have ( u, d ) = ( c, d ) = 1.Put t ( x, y ) = ( − ( x − y − / for x, y ∈ Z + 1. If a prime p satisfies p | c ,we have h c, d i p = (cid:18) dp (cid:19) ord p c p ≥ t ( u, d ) (cid:18) | d | (cid:19) ord c p = 2 . If c is odd, then we have (cid:18) dc (cid:19) ∗ = Q p | c (cid:18) dp (cid:19) ord p c = v ( g ). If c is even, thenwe have (cid:18) u | d | (cid:19) (cid:18) d | u | (cid:19) = t ( c, d ) t ( u, d ) (see [11, p.51]). Thus we have (cid:16) cd (cid:17) ∗ = t ( c, d ) (cid:18) | d | (cid:19) ord c (cid:18) u | d | (cid:19) = t ( u, d ) (cid:18) | d | (cid:19) ord c (cid:18) d | u | (cid:19) = v ( g ) . (cid:3) Put K f = Y v< ∞ SL ( o v ) . Then K f is a compact open group of SL ( A f ). The inverse image of K f in ^ SL ( A f ) is denoted by ˜ K f . We have SL ( o ) = SL ( F ) ∩ K f · SL ( F ∞ ). Proposition 3.
Let v be a multiplier system of half-integral weight forSL ( o ). Then there exists a genuine character λ : ˜ K f → C × such that v λ = v . N MULTIPLIER SYSTEMS AND THETA FUNCTIONS 17
Proof. If F = Q , the assertion is proved by Proposition 2. If F = Q , let v η be the multiplier system of η ( z ) in (3). PutΓ(12) = (cid:26)(cid:18) a bc d (cid:19) ∈ SL ( Z ) | a ≡ d ≡ , b ≡ c ≡ (cid:27) . By Lemma 12, we have v η ( γ ) = v ( γ ) for γ ∈ Γ(12). Since v η ( γ ) / v ( γ ) = 1for any γ ∈ D (SL ( Z )), we have v ( γ ) = v ( γ ) for any γ ∈ D (SL ( Z )) ∩ Γ(12),which is a congruence subgroup. By Lemma 11, there exists a genuinecharacter λ : ˜ K f → C × such that v λ = v . (cid:3) Corollary 2.
There exists a multiplier system v of half-integral weight forSL ( o ) if and only if 2 splits completely in F/ Q . There exists a genuinecharacter of ^ SL ( o v ) for any v < ∞ , provided that this condition holds. Proposition 4.
Suppose that 2 splits completely in F/ Q . Let v λ be amultiplier system of half-integral weight of SL ( o ), where λ = Q v< ∞ λ v is agenuine character of ˜ K f . Put S = { v < ∞ | F = Q } and T = { v < ∞ | q v = 3 } . If q v is odd, let ǫ v be a genuine character of SL ( o v ) in Lemma4. We set S = { v ∈ T | λ v = ǫ v } . Let β v be a element of F × such that λ v = µ β v for v ∈ S ∪ S . Then we have v λ ( γ ) = v ( γ ) Y v ∈ S ∪ S κ v ( β v , ι v ( γ )) γ = (cid:18) a bc d (cid:19) ∈ SL ( o ) . Here, if v ∈ S , κ v ( β v , g ) = ( ψ β v ( − ( a + d ) c + 3 c ) c ∈ Z × ψ β v (( c − b ) d − d − c ∈ Z and if v ∈ S , κ v ( β v , g ) = ψ β v ( − ( a + d ) c + bd ( c − g = (cid:18) a bc d (cid:19) ∈ SL ( o v ). Note that κ v ( β v , ι v ( γ )) is a continuous functionon γ . Proof.
We have v λ ( γ ) = λ (˜ ι f ( γ )) = Q v< ∞ λ v ([ ι v ( γ )]). If v / ∈ S ∪ S , thenwe have ǫ v ([ g ]) = s v ( g ) for any g ∈ SL ( o v ). If v ∈ S (resp. S ), we have µ β v ([ g ]) = s v ( g ) κ v ( β v , g ) by (9) (resp. (10)) for any g ∈ SL ( o v ). Thisproves the proposition. (cid:3) The condition of the existence of a theta function
Suppose that 2 splits completely in F/ Q . By Lemma 3, there exists agenuine character λ v : ^ SL ( o v ) → C × for any v < ∞ . If v < ∞ , put K v = SL ( o v ). If v | ∞ , put K v = SO(2). Then K v is a maximal compactsubgroup of SL ( F v ) for any v . Let β be an element of F × and ψ β the char-acter of A /F as in Section 1. For any v , we denote the Weil representationof ^ SL ( F v ) by ω ψ β ,v . Let {∞ , · · · , ∞ n } be the set of v | ∞ and S ( R ) the Schwartz space of R .We have an irreducible decomposition ω ψ β ,v = ω + ψ β ,v ⊕ ω − ψ β ,v , where ω + ψ β ,v (resp. ω − ψ β ,v ) is an irreducible representation of the set of even(resp. odd) functions in S ( R ) (see [14, Lemma 2.4.4]).The group ^ SL ( R ) has a maximal compact subgroup ^ SO(2), which isthe inverse image of SO(2) in ^ SL ( R ). It is known that if λ v : ^ SO(2) → C × is a genuine character, dim C ( ω ψ β ,v , S ( R )) λ v is at most 1. Let λ ∞ , / be a genuine character of lowest weight 1/2 with respect to ( ω + ψ β ,v , S ( R ))and λ ∞ , / of lowest weight 3/2 with respect to ( ω − ψ β ,v , S ( R )). For β > ω + ψ β ,v , S ( R )) λ ∞ , / = C e ( iι v ( β ) x ) and ( ω − ψ β ,v , S ( R )) λ ∞ , / = C xe ( iι v ( β ) x )are spaces of lowest weight vectors. If β <
0, there exist no lowest weightvectors with respect to ( ω + ψ β ,v , S ( R )) or ( ω − ψ β ,v , S ( R )).Note that λ v ( s v (SL ( o v ))) = 1 for any v < ∞ except for finitely manyplaces. Then a genuine character λ f : ˜ K f → C × is given by λ f ( g ) = Q v< ∞ λ v ( g v ) for g = ( g v ) v ∈ ˜ K f . Put w = ( w , · · · , w n ) ∈ { / , / } n . Wedefine an automorphy factor j λ f ,w ( γ, z ) for γ ∈ SL ( o ) and z = ( z , · · · , z n ) ∈ h n by j λ f ,w ( γ, z ) = Y v< ∞ λ v ([ ι v ( γ )]) n Y i =1 ˜ j ([ ι i ( γ )] , z i ) w i . In particular, we have j λ f ,w ( − , z ) = Q v< ∞ λ v ([ − ]) × ( − P w i . If itdoes not equal 1, the space of Hilbert modular forms of weight w for SL ( o )is { } .Put K = K f × Q v |∞ SO(2). There exists a genuine character λ : ˜ K → C × such that its v -component equals λ v , where λ ∞ i is λ ∞ , / or λ ∞ , / for1 ≤ i ≤ n . Then we have an automorphy factor j λ f ,w ( γ, z ) correspondingto λ such that λ ∞ i = λ ∞ ,w i .Let M w (SL ( o ) , λ f ) be the space of Hilbert modular forms on h n withrespect to j λ f ,w ( γ, z ). A holomorphic function h ( z ) of h n belongs to thespace M w (SL ( o ) , λ f ) if and only if h ( γ ( z )) = j λ f ,w ( γ, z ) h ( z ) , where γ ( z ) = ( ι ( γ )( z ) , · · · , ι n ( γ )( z n )) for γ ∈ SL ( o ) and z ∈ h n . (When F = Q , the usual cusp condition is also required.)For each g ∈ ^ SL ( A ), there exist γ ∈ SL ( F ), g ∞ ∈ ^ SL ( R ) n and g f ∈ ˜ K f such that g = γg ∞ g f by the strong approximation theorem for SL ( A ). Put i = ( √− , · · · , √− ∈ h n . For h ∈ M w (SL ( o ) , λ f ), put ϕ h ( g ) = h ( g ∞ ( i )) λ f ( g f ) − n Y i =1 ˜ j ( g ∞ i , √− − w i . Then ϕ h is an automorphic form on SL ( F ) \ ^ SL ( A ). N MULTIPLIER SYSTEMS AND THETA FUNCTIONS 19
Let A w (SL ( F ) \ ^ SL ( A ) , λ f ) be the space of automorphic forms ϕ onSL ( F ) \ ^ SL ( A ) satisfying the following conditions (1), (2), and (3).(1) ϕ ( gk ∞ ) = ϕ ( g ) Q ni =1 ˜ j ( k ∞ ,i , √− − w i for any g ∈ ^ SL ( A ) and k ∞ =( k ∞ , , . . . , k ∞ ,n ) ∈ ^ SO(2) n .(2) ϕ is a lowest weight vector with respect to the right translation of ^ SL ( R ) n .(3) ϕ ( gk ) = λ f ( k ) − ϕ ( g ) for any g ∈ ^ SL ( A ) and k ∈ ˜ K f .Then Φ : h ϕ h gives rise to an isomorphism M w (SL ( o ) , λ f ) −→∼ A w (SL ( F ) \ ^ SL ( A ) , λ f ) . For ϕ ∈ A w (SL ( F ) \ ^ SL ( A ) , λ f ), put h = Φ − ( ϕ ). Then we have h ( z ) = ϕ ( g ∞ ) n Y i =1 ˜ j ( g ∞ i , √− w i , g ∞ ∈ ^ SL ( R ) n , g ∞ ( i ) = z. Now suppose that v < ∞ . Let d be the different of F/ Q and q v the orderof the residue field o v / p v . When q v is odd, let ǫ v be a genuine character ǫ F in Lemma 4. Put S = { v | F v = Q } , T = { v < ∞ | q v = 3 } and S = { v ∈ T | λ v = ǫ v } . Since 2 splits completely in F/ Q , we have | S | = n . By Lemma 7 and Lemma 8, ( ω ψ β ,v , S ( F v )) λ v is not 0 if and onlyif we have ord v ψ β,v ≡ ( λ v = ǫ v ω ψ β ,v , S ( F v )) λ v = 0 for any v < ∞ , there exists a fractionalideal a such that(16) (8 β ) d Y v ∈ S p v = a . The set of totally positive elements of F is denoted by F × + . Replacing β with βγ and a with ( a γ ) in (16) for γ ∈ F × + , we may assume ord v a = 0 for v ∈ S ∪ S . Then we have ord v ψ β,v = − −
3) for v ∈ S (resp. S ).Conversely, suppose that there exists a fractional ideal a satisfying (16)for a subset S ⊂ T . For v < ∞ , put λ v = ( ǫ v if ord v ψ β,v ≡ µ β if ord v ψ β,v ≡ , where µ β is a genuine character in Lemma 5 or Lemma 6. By Lemma 7 andLemma 8, we have ( ω ψ β ,v , S ( F v )) λ v = 0 for any v < ∞ . Let λ : ˜ K → C × bea genuine character such that its v -component equals λ v , where λ ∞ i = λ ∞ ,w i for w i ∈ { / , / } . Put S ∞ = {∞ i | w i = 3 / } .From now on, suppose that β ∈ F × + . Let S ( A ) be the Schwartz spaceof A and ( ω ψ β , S ( A )) λ the set of functions φ = Q v φ v ∈ S ( A ) such that φ v ∈ ( ω ψ β ,v , S ( F v )) λ v for any v . For φ ∈ S ( A ), we define the theta functionΘ φ by(17) Θ φ ( g ) = X ξ ∈ F ω ψ β ( g ) φ ( ξ ) g = ( g v ) ∈ ^ SL ( A ) , where ω ψ β ( g ) φ ( ξ ) = Q v ω ψ β ,v ( g v ) φ v ( ι v ( ξ )) is essentially a finite product.We have Θ φ ( gk ) = λ ( k ) − Θ φ ( g ) for any g ∈ ^ SL ( A ) and k ∈ ˜ K f . If φ ∈ ( ω ψ β , S ( A )) λ , then Θ φ is a Hilbert modular form of weight w = ( w , · · · , w n ).It is known that ω ψ β = M S ω ψ β ,S , ω ψ β ,S = O v ∈ S ω − ψ β ,v ! ⊗ O v / ∈ S ω + ψ β ,v ! , where S ranges over all finite subsets of places of F (see [5, § ω ψ β to the space of automorphic forms on ^ SL ( A ) byΘ( φ )( g ) = Θ φ ( g ). Then it is known that(18) Im(Θ) ≃ M | S | :even ω ψ β ,S , (see [5, Proposition 3.1]).Let G be the set of triplets ( β, S , a ) of β ∈ F × + , a subset S ⊂ T and afractional ideal a of F satisfying (16) and the condition (A),(A) | S | + | S | + | S ∞ | ∈ Z . We define an equivalence relation ∼ on G by( β, S , a ) ∼ ( β ′ , S ′ , a ′ ) ⇐⇒ S = S ′ , β ′ = γ β, a ′ = γ a for some γ ∈ F × . Theorem 1.
Suppose that 2 splits completely in F/ Q . Let β ∈ F × + , λ :˜ K → C × and w , . . . , w n ∈ { / , / } be as above. Then there exists φ = Q v φ v ∈ ( ω ψ β , S ( A )) λ such that Θ φ = 0 if and only if there exists afractional ideal a of F such that ( β, S , a ) ∈ G . Proof.
Let λ v : ^ SL ( o v ) → C × be the v -component of λ for any v < ∞ . We already proved that there exists Q v< ∞ φ v = 0 such that φ v ∈ ( ω ψ β ,v , S ( F v )) λ v for any v < ∞ if and only if there exists a fractionalideal a of F satisfying (16). Suppose that the equivalent conditions hold.Since we have ( ω + ψ β ,v , S ( R )) λ ∞ , / = C e ( iι v ( β ) x ) and ( ω − ψ β ,v , S ( R )) λ ∞ , / = C xe ( iι v ( β ) x ) for any v | ∞ , there exists a nonzero φ = Q v φ v ∈ ( ω ψ β , S ( A )) λ .It is clear that if there exists a nonzero φ = Q v φ v ∈ ( ω ψ β , S ( A )) λ , Q v< ∞ φ v =0 satisfies φ v ∈ ( ω ψ β ,v , S ( F v )) λ v for any v < ∞ .Suppose there exists a nonzero φ = Q v φ v ∈ ( ω ψ β , S ( A )) λ . Note that | S | + | S | + | S ∞ | is the number of v such that φ v is an odd function. Then | S | in (18) is | S | + | S | + | S ∞ | . By (18), it is clear that Θ φ = 0 if and onlyif the condition (A) holds. (cid:3) N MULTIPLIER SYSTEMS AND THETA FUNCTIONS 21
Let H be a group of fractional ideals that consists of all elements of theform Y v ∈ T p e v v , X v e v ∈ Z . Let Cl + be the narrow ideal class group of F . Put Cl +2 = { c | c ∈ Cl + } .We denote the image of the group H (resp. b ∈ Cl + ) in Cl + / Cl +2 by ¯ H (resp. [ b ]). Theorem 2.
Suppose that 2 splits completely in F/ Q . Let w , . . . , w n ∈{ / , / } be as above.(1) Suppose that | S | + | S ∞ | is even. Then there exists ( β, S , a ) ∈ G ifand only if [ d ] ∈ ¯ H .(2) Suppose that | S | + | S ∞ | is odd. Then there exists ( β, S , a ) ∈ G ifand only if T = ∅ and [ dp v ] ∈ ¯ H . Here, v is any fixed element of T . Proof.
We prove the theorem in case (1). The proof for case (2) is similar.If [ d ] ∈ ¯ H , we have (8 β ) d Q v ∈ T p e v v = a ′ such that P v e v is even fora fractional ideal a ′ and β ∈ F × + . Put S = { v ∈ T | e v : odd } . Since | S | + | S | + | S ∞ | is even, we have ( β, S , a ) ∈ G , where a = Y v ∈ T \ S p − e v / v a ′ . Conversely, if there exists ( β, S , a ) ∈ G , it satisfies (16) and | S | is even.Then we have [ d ] = Q v ∈ S [ p v ] ∈ ¯ H . (cid:3) Let w i be 1/2 or 3/2 for 1 ≤ i ≤ n . Suppose that there exists ( β, S , a ) ∈ G . Replacing ( β, S , a ) with an equivalent element of G , we may assumeord v a = 0 for v ∈ S ∪ S . Let f v be the function f in (8) and put f = Y v ∈ S ∪ S f v × Y v< ∞ ,v / ∈ S ∪ S ch a − v , where a v = ao v . Put φ = f × Q ni =1 f ∞ ,i , where f ∞ ,i ( x ) = x w i − (1 / e ( iι i ( β ) x )for x ∈ R . By Theorem 1, there exists Θ φ = 0 of weight w = ( w , · · · , w n ).Put z = ( z , · · · , z n ) , i = ( √− , · · · , √− ∈ h n . We define x i , y i ∈ R by z i = x i + √− y i for 1 ≤ i ≤ n . Then we have z = g ∞ ( i ), where g ∞ = ( g ∞ , · · · , g ∞ n ) ∈ SL ( R ) n , g ∞ i = y / i y / i x i y − / i ! . Since λ v ([1 ]) = 1for v < ∞ , we haveΘ φ ( g ∞ ) = X ξ ∈ a − f ( ι f ( ξ )) n Y i =1 ω ψ β , ∞ i ([ g ∞ i ]) f ∞ ,i ( ι i ( ξ )) . Theorem 3.
Let φ and Θ φ be as above. We define a theta function θ φ : h n → C by θ φ ( z ) = X ξ ∈ a − f ( ι f ( ξ )) Y ∞ i ∈ S ∞ ι i ( ξ ) n Y i =1 e ( z i ι i ( βξ )) . Then θ φ is a nonzero Hilbert modular form of weight w for SL ( o ) withrespect to a multiplier system.Every theta function of weight w for SL ( o ) with a multiplier system maybe obtained in this way. Proof.
Since ω ψ β , ∞ i ([ g ∞ ,i ]) f ∞ ,i ( ι i ( ξ )) = y w i / i ι i ( ξ ) w i − (1 / e ( z i ι i ( βξ )) , we have θ φ ( z ) = Θ φ ( g ∞ ) × Q ni =1 y − w i / i . Then θ φ is nonzero. Note that˜ j ([ g ∞ i ] , √− w i = y − w i / i . Since φ ∈ ( ω ψ β , S ( A )) λ , we have Θ φ ∈ A w (SL ( F ) \ ^ SL ( A ) , λ f ). Then wehave θ φ = Φ − (Θ φ ) ∈ M w (SL ( o ) , λ f ). The multiplier system of θ φ is v λ given by v λ ( γ ) = v ( γ ) Y v ∈ S ∪ S κ v ( β, γ ) γ ∈ SL ( o ) , where κ v for v ∈ S ∪ S is the function in Proposition 4.By Proposition 3, if θ is a theta function of weight w for SL ( o ) with amultiplier system v , we have a genuine character λ f of ˜ K f such that v = v λ f .Let λ = λ f × Q ni =1 λ ∞ ,w i be a genuine character of ˜ K . Then there existsnonzero φ ∈ ( ω ψ β , S ( A )) λ such that θ = θ φ up to constant, which completesthe proof. (cid:3) Proposition 5.
Let Cl be the usual ideal class group of F . Let Sq : Cl → Cl + be the homomorphism given by [ a ] [ a ] for a fractional ideal a of F .The number of equivalence classes of G is equal to[ E + : E ] X S ⊂ T | Sq − ([ d Y v ∈ S p v ]) | , where S ranges over all subset of T satisfying (A). Here, E + is the groupof totally positive units of F and E is the subgroup of squares of units of F . Proof.
We follow the argument of Hammond [8] Theorem 2.9. For given S satisfying (A), the number of ideal classes [ a ] such that a is narrowlyequivalent to d Q v ∈ S p v is equal to | Sq − ([ d Q v ∈ S p v ]) | . Then for a givenfractional ideal a such that a is narrowly equivalent to d Q v ∈ S p v , thenumber of equivalence classes of triplets of the form ( β, S , a ) such that β ∈ F × + satisfying (16) is equal to [ E + : E ]. (cid:3) The case F = Q or F is a real quadratic field Suppose that F = Q . If S ∞ = ∅ , the equivalence class of G is { (1 / , { } , Z ) } .The theta function obtained by { (1 / , { } , Z ) } equals 2 η ( z ). Then its mul-tiplier system equals v η in (3). If S ∞ = {∞} , then the equivalence classof G is { (1 / , ∅ , Z ) } . The theta function obtained by { (1 / , ∅ , Z ) } equals2 η ( z ). Then its multiplier system equals the cubic power of v η . N MULTIPLIER SYSTEMS AND THETA FUNCTIONS 23
Now suppose that F = Q ( √ D ), where D > β, S , a ) ∈ G , one of the followings holds.(C1) (8 β ) d = a and S = ∅ .(C2) (8 β ) dp = a such that N F/ Q ( p ) = 3 and S = { p } .(C3) (8 β ) dp ¯ p = a such that N F/ Q ( p ) = N F/ Q (¯ p ) = 3 and S = { p , ¯ p } .If | S ∞ | is even, (C1) or (C3) holds. If | S ∞ | is odd, (C2) holds.Suppose that D ≡ F/ Q and we have d = ( √ D ). Lemma 13.
Let N be a positive square-free integer. Put L = Q ( √− L = Q ( √− N is a norm of an element of L × .(b) N is a norm of an integer of L .(c) No prime factor of N are inert in L/ Q .(d) There exist integers u and v such that N = u + v (resp. N =3 u + v ). Proof.
The statements (b), (c) and (d) are equivalent by [3, §
68 and § L = Q ( √− §
68] treated the case N is odd but the proofis valid in general case. If L = Q ( √− §
70] treated the case N is oddand not divisible by 3, but the proof is valid in general case. If (b) holds,then clearly (a) holds.It suffices to show that if (a) holds, then (c) holds. Suppose that α N ∈ L × satisfies N = N L/ Q ( α N ). If a prime p is inert in L/ Q , it is a prime elementof L and we have N L/ Q ( p ) = p . Then if p | N , we have p | α N and p | N ,which contradicts that N is square-free. (cid:3) We consider an analogy of the following: if K is a real quadratic field,then a necessary and sufficient condition that the narrow ideal class of thedifferent of K be a square is that the discriminant D of K be the sum oftwo integer squares (see [8] Proposition 3.1). Lemma 14.
A necessary and sufficient condition that the narrow ideal classof dp is a square for a prime ideal p which has norm 3 is that D is of theform 3 u + v for some u, v ∈ N . Proof.
Suppose that the narrow ideal class of dp is a square with N F/ Q ( p ) =3. Then there exists σ ∈ F × + and a fractional ideal a of F such that ( σ ) dp = a . Taking the norm of both sides, we have 3 N F/ Q ( σ ) D = A , where A is the norm of a . Put σ = s + t √ D for s, t ∈ Q such that s >
0. Since3( s − t D ) D = A , we have D = (cid:18) tDs (cid:19) + 3 (cid:18) A s (cid:19) . Put L = Q ( √− D ∈ N L/ Q ( L × ). Lemma 13 implies that D = 3 u + v for some u, v ∈ N .We assume that there exists u, v ∈ N such that D = 3 u + v . Since D ≡ u ′ = u/ ∈ Z . Put ρ = ( v + √ D ) /
2. Then we have N F/ Q ( ρ ) = ( v − D ) / − u ′ . Let q = q Q be a prime ideal which divides ρ , where Q is a rational prime which is divisible by q . Since ρ/Q / ∈ o , wehave N F/ Q ( q ) = Q and ord q ρ =ord Q u ′ . Therefore if Q = 3, ord q ρ is even.Since N F/ Q ( ρ ) = − u ′ , there exists a prime ideal q of F which dividesboth 3 and ρ . Since 3 splits or ramifies in F/ Q , ord q ρ is odd. Put(19) a = Y q ∤ q (ord q ρ ) / × q (ord q ρ +1) / . Then we have ( √ Dρ ) q = da . Since √ Dρ ∈ F × + , we have dq = ( da ) inCl + . (cid:3) Proposition 6.
Suppose that F = Q ( √ D ), where D > D ≡ β ∈ F × + and a fractional ideal a satisfying (C1) if andonly if p ≡ p | D .(2) There exist β ∈ F × + and a fractional ideal a satisfying (C2) if andonly if p ≡ p | D .(3) There exists β ∈ F × + and a fractional ideal a satisfying (C3) if andonly if D ≡ p ≡ p | D . Proof.
For a prime ideal p such that N F/ Q ( p ) = 3, the equation (8 β ) dp = a implies that the narrow ideal class of dp is a square. Note that a positiveinteger x is of the form 3 u + v for some u, v ∈ N if and only if any prime p which divides x satisfies p ≡ β ) d = a implies that the narrow ideal class of d is asquare. Note that a positive integer x is of the form u + v for some u, v ∈ N if and only if any prime p which divides x satisfies p ≡ p and ¯ p such that such that N F/ Q ( p ) = N F/ Q (¯ p ) = 3 if and only if 3 splits in F/ Q . This condition holds if and onlyif D ≡ D ≡ p ¯ p = (3). Then theequation (8 β ) dp ¯ p = a implies that the narrow ideal class of d is a square.Thus, similarly to the first assertion, [8] Proposition 3.1 proves the thirdassertion. (cid:3) Example: put D = 793 = 13 ·
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