Irreducible representations and Artin L-functions of quasi-cyclotomic fields
aa r X i v : . [ m a t h . N T ] A p r Irreducible representations and Artin L -functions of quasi-cyclotomic fields Sunghan Bae (1) , Yong Hu (2) , Linsheng Yin (2) (1) Department of Mathematics, KAIST, Daejeon, Korea(2) Dept. of Math. Sciences, Tsinghua Univ., Beijing 100084, ChinaNovember 21, 2018
Abstract
We determine all irreducible representations of primary quasi-cyclotomicfields in this paper. The methods can be applied to determine the irre-ducible representations of any quasi-cyclotomic field. We also computethe Artin L -functions for a class of quasi-cyclotomic fields. A quadratic extension of a cyclotomic field, which is non-abelian Galois overthe rational number field Q , is called a quasi-cyclotomic field. All quasi-cyclotomic fields are described explicitly in [8] followed the works in [1] and[3]. They are generated by a canonical Z / Z -basis. The minimal quasi-cyclotomic field containing the quadratic roots of one element of the ba-sis is called a primary quasi-cyclotomic field. L.Yin and C.Zhang [7] havestudied the arithmetic of any quasi-cyclotomic field. In this paper we deter-mine all irreducible representations of primary quasi-cyclotomic fields. Themethods apply to determine the irreducible representations of an arbitraryquasi-cyclotomic field. We also compute the Artin L -functions for a class ofquasi-cyclotomic fields.First we recall the constructions of primary quasi-cyclotomic fields. Let S be the set consisting of − p ∈ S , we put¯ p = 4 , , p and set p ∗ = − , , ( − p − p if p = − , K = Q ( ζ ¯ pq ) be the cyclotomic field of conductor¯ pq . For a class [ a ] ∈ Q / Z , we set sin[ a ] = 2 sin aπ for 0 < a < p < q , we define v pq = p − Y i =0 q − Y j =0 sin[ iq + jpq ]sin[ jp + ipq ] ( p > v q = sin[ ]sin[ q ] q − Y j =0 sin[ j q ] · sin[ j − q ]sin[ j + i q ] · sin[ jq ] · sin[ j − q ] . For p < q ∈ S , we put u pq := √ q ∗ if p = − v pq if p = 2 or p ≡ q ≡ √ p · v pq if p ≡ , q ≡ √ q · v pq if p ≡ , q ≡ √ pq · v pq if p ≡ q ≡ . Let ˜ K = K ( √ u pq ). Then ˜ K is the minimal one in all quasi-cyclotomicfields which contain √ u pq . We call these fields ˜ K primary quasi-cyclotomicfields. Let G = Gal( K/ Q ) and ˜ G = Gal( ˜ K/ Q ). We always denote by ǫ the unique non-trivial element of Gal( ˜ K/K ). If ( p, q ) = ( − , G is generated by two elements σ − and σ , where σ − ( ζ ) = ζ − and σ ( ζ ) = ζ . If p >
2, then G is generated by two elements σ p and σ q ,where σ p ( ζ p ) = ζ ap , σ p ( ζ q ) = ζ q and σ q ( ζ p ) = ζ p , σ q ( ζ q ) = ζ bq , with a, b beinggenerators of ( Z /p Z ) ∗ and ( Z /q Z ) ∗ respectively. If p = 2, then G is generatedby three elements σ − , σ and σ q , where σ − , σ act on ζ as above and on ζ q trivially, and σ q acts on ζ q as above and on ζ trivially.Next we describe the group ˜ G by generators and relations. An element σ ∈ G has two liftings in ˜ G . By [Sect.3, 7] the action of the two liftings on √ u pq has the form ± α √ u pq or ± α √ u pq / √− α >
0. We fix the lifting˜ σ of σ to be the one with the positive sign. Then the other lifting of σ is˜ σǫ . The group ˜ G is generated by ǫ, ˜ σ p and ˜ σ q (and ˜ σ − if p = 2). Clearly ǫ commutes with the other generators. In addition, we have ˜ σ p ˜ σ q = ˜ σ q ˜ σ p ǫ (and ˜ σ − commutes with ˜ σ and ˜ σ q if p = 2). For an element g of a group,we denote by | g | the order of g in the group. Let log − : {± } → Z / Z bethe unique isomorphism. For an odd prime number p and an integer a with p ∤ a , let ( ap ) be the quadratic residue symbol. We also define ( a ) = ( a − ) = 1for any a . Then we have, see [7, Th.3], | ˜ σ p | = (1 + log − ( q ∗ p )) | σ p | and | ˜ σ q | = (1 + log − ( p ∗ q )) | σ q | , with the exception that ˜ σ = 2 | σ | when ( p, q ) = ( − , p = 2, wehave furthermore | ˜ σ − | = | σ − | . Thus we have determined the group ˜ G bygenerators and relations. 2 Abelian subgroup of index 2
In this section we construct a subgroup of ˜ G of index 2 and determine thestructure of the subgroup. We consider the following three cases separately:Case A: | e σ p | = | σ p | and | e σ q | = | σ q | ;Case B: | e σ p | = 2 | σ p | , | e σ q | = | σ q | or | e σ p | = | σ p | , | e σ q | = 2 | σ q | ;Case C: | e σ p | = 2 | σ p | and | e σ q | = 2 | σ q | .All the three cases may happen. In fact, the case (A) happens if and onlyif ( p ∗ q ) = ( q ∗ p ) = 1; the case (B) happens if and only if ( p ∗ q ) = ( q ∗ p ) or( p, q ) = ( − , p ∗ q ) = ( q ∗ p ) = − N of e G to be N = ( < e σ − , e σ , e σ q , ε > if p = 2 < e σ p , e σ q , ε > if p = 2 ( A . N is abelian of index 2 and is a directsum of the cyclic groups generated by the elements. Thus we have N ∼ = Z / Z ⊕ Z / (( q − / Z ⊕ Z / Z if p = − Z / Z ⊕ Z / Z ⊕ Z / (( q − / Z ⊕ Z / Z if p = 2 Z / ( p − Z ⊕ Z / (( q − / Z ⊕ Z / Z if p > . ( A . N of e G to be N = < e σ − , e σ , e σ q > if p = 2 < e σ p , e σ q > if p = 2 and | e σ q | = 2 | σ q | < e σ p , e σ q > if | e σ p | = 2 | σ p | . ( B . N is abelian and has index 2 in e G . In addition, we have N ∼ = Z / Z ⊕ Z / Z if ( p, q ) = ( − , Z / Z ⊕ Z / ( q − Z if p = − , q > Z / Z ⊕ Z / Z ⊕ Z / ( q − Z if p = 2 Z / ( p − Z ⊕ Z / ( q − Z if p > . ( B . p, q are odd prime numbers. Let v ( p − p −
1. We define the subgroup N of e G to be N = ( < e σ p , e σ q > if v ( p − ≤ v ( q − < e σ p , e σ q > if v ( p − > v ( q − . ( C . N is an abelian subgroup of e G . When v ( p − ≤ v ( q − | N | = | e σ p | · | e σ q || < e σ p > ∩ < e σ q > | = ( p − · q − . So [ e G : N ] = 2 and N is a normal subgroup of e G . We have the sameresult when v ( p − > v ( q − < e σ p , e σ q > is alwaysan abelian subgroup of e G of index 2. But we can not get all irreduciblerepresentations from the inducement of the representations of this subgroupwhen v ( p − > v ( q − N in two cases.Next we determine the structure of the subgroup N in the case C. Weconsider the case v ( p − ≤ v ( q −
1) in detail. Let d = gcd( p − , q − s = ( p − / d and t = ( q − /d . Choose u, v ∈ Z such that us + vt = 1.We have the relations( e σ p ) p − = 1 , ( e σ p ) p − = ε = e σ q − q . Let M be the free abelian group generated by two words α , β . Let α = ( p − α ; β = p − α − ( q − β ;and let M be the subgroup of M generated by α , β . Then M is the kernelof the homomorphism M −→ N ; α e σ p , β e σ q . So we have N ∼ = M/M . Define the matrix A := (cid:18) p − p − − q (cid:19) . Then ( α , β ) = ( α , β ) · A . We determine the structure of M by consideringthe standard form of A . Define P := (cid:18) u v − t s (cid:19) ∈ SL ( Z ) ; Q := (cid:18) tv − − − tv + 2 (cid:19) ∈ SL ( Z ) . Then B := P AQ = (cid:18) d − s ( q − (cid:19) is the standard form of A . Let( τ , µ ) = ( α , β ) P − and ( τ , µ ) = ( α , β ) Q. τ , µ ) = ( τ , µ ) B , M = Z τ ⊕ Z µ and M = Z dτ ⊕ Z s ( q − µ . Wethus have N ∼ = M/M ∼ = Z /d Z ⊕ Z / s ( q − Z . By abuse of notation, we also write( τ , µ ) = ( e σ p , e σ q ) P − = ( e σ sp e σ tq , e σ − vp e σ uq ) . Then τ , µ are of order d , s ( q −
1) respectively, and N is a direct sum of < τ > and < µ > . We have e σ p = τ u µ − t and e σ q = τ v µ s . When v ( p − > v ( q − N in the same way. So in the case (C) we have N ∼ = ( Z /d Z ⊕ Z / s ( q − Z if v ( p − ≤ v ( q − Z /d ′ Z ⊕ Z / s ′ ( p − Z if v ( p − > v ( q − , ( C . d = gcd( p − , q − , s = ( p − / d and d ′ = gcd( p − , q − ) , s ′ =( q − / d ′ .Now we summarize our results in the following Proposition 2.1.
The abelian subgroup N of the group e G of index 2 definedin (A2.1), (B2.1) and (C2.1) has the structure described in (A2.2), (B2.2)and (C2.2) in the cases (A), (B) and (C), respectively. In particular, everyirreducible representation of e G has dimension 1 or 2. We determine all irreducible representations of e G in this section. We will usesome basic facts from representation theory freely. For the details, see [6].It is well-known that the 1-dimensional representations of e G correspondbijectively to those of the maximal abelian quotient G of e G , which are Dirich-let characters. So we mainly construct the 2-dimensional irreducible repre-sentations of e G . From the dimension formula of all irreducible representa-tions, we see that e G has | G | / N be the subgroup of e G defined in last section. Let e G = N ∪ σN be a decomposition of cosets. If ρ : N → C ∗ is a representationof N , the induced representation e ρ of ρ is a representation of e G of dimension2. The space of the representation e ρ is V = Ind e GN ( C ) = C [ e G ] ⊗ C [ N ] C withbasis e = 1 ⊗ e = σ ⊗
1. The group homomorphism e ρ : e G −→ GL( V ) ≃ GL ( C )is given under the basis by e ρ ( e σ ) = (cid:18) ρ ( e σ ) ρ ( e σσ ) ρ ( σ − e σ ) ρ ( σ − e σσ ) (cid:19) , ∀ e σ ∈ e G, (3.1)5here ρ ( e σ ) = 0 if e σ / ∈ N . The representation e ρ is irreducible if and only if ρ = ρ τ for every τ ∈ e G \ N , where ρ τ is the conjugate representation of ρ defined by ρ τ ( x ) = ρ ( τ − xτ ) , ∀ x ∈ N .
Since N is abelian, we only need to check ρ = ρ σ .Now we begin to construct all 2-dimensional irreducible representationsof e G . As in last section, we consider the three cases separately. In addition,we consider the case when p and q are odd prime numbers in details, andonly state the results in the cases when p = − Assume p >
2. We have in this case N = h e σ p , e σ q , ε i and N ∼ = Z / ( p − Z ⊕ Z / (( q − / Z ⊕ Z / Z . Every irreducible representation of N can be written as ρ ijk : N −→ C ∗ with ρ ijk ( e σ p ) = ζ ip − ; ρ ijk ( e σ q ) = ζ jq − ; ρ ijk ( ε ) = ( − k . where 0 ≤ i < p − , ≤ j < q − and k = 0 ,
1. Since e G = N ∪ e σ q N and ρ e σ q ijk ( e σ p ) = ρ ijk ( ε ) ρ ijk ( e σ p ) = ( − k ρ ijk ( e σ p ), we have ρ e σ q ijk = ρ ijk ⇐⇒ k = 1 . Write ρ ij = ρ ij . The induced representation e ρ ij : e G −→ GL ( C ) of ρ ij isgiven by e ρ ij ( e σ p ) = (cid:18) ζ ip − − ζ ip − (cid:19) , e ρ ij ( e σ q ) = (cid:18) ζ jq − (cid:19) , e ρ ij ( ε ) = − I, ( A . I is the identity matrix of degree 2. Since e ρ ij ( e σ p ) = (cid:18) ζ ip − ζ ip − (cid:19) and e ρ ij ( e σ q ) = (cid:18) ζ jq − ζ jq − (cid:19) , we see that the representations e ρ ij with 0 ≤ i < p − , ≤ j < q − are irre-ducible and are not isomorphic to each other, by considering the values of thecharacters of these representations at e σ p and e σ q . The number of these rep-resentations is p − · q − = | G | . So they are all the irreducible representationsof e G of dimension 2.Similarly, when p = −
1, all irreducible representations of e G of dimension2 are e ρ j with 0 ≤ j < q − , where e ρ j ( e σ − ) = (cid:18) − (cid:19) , e ρ j ( e σ q ) = (cid:18) ζ jq − (cid:19) , e ρ ( ε ) = − I ( A . p = 2, all irreducible representations of e G of dimension 2 are ¯ ρ ij with 0 ≤ i ≤ ≤ j < q − , where ¯ ρ ij ( ε ) = − I and¯ ρ ij ( e σ − ) = ( − i I, ¯ ρ ij ( e σ ) = (cid:18) − (cid:19) , ¯ ρ ij ( e σ q ) = (cid:18) ζ jq − (cid:19) . ( A . Assume p > | e σ q | = 2 | σ q | . Then N = h e σ p , e σ q i , and N ∼ = Z / ( p − Z ⊕ Z / ( q − Z . Any irreducible representation of N has the form ρ ij : N −→ C ∗ , where ρ ij ( e σ p ) = ζ ip − , ρ ij ( e σ q ) = ζ jq − , ρ ij ( ε ) = ρ ij ( e σ q ) q − = ( − j , and 0 ≤ i < p − , ≤ j < q −
1. It is easy to check that ρ e σ q ij = ρ ij ⇐⇒ j ≡ . The induced representation e ρ ij : e G −→ GL ( C ) of ρ ij with odd j is given by e ρ ij ( e σ p ) = (cid:18) ζ ip − − ζ ip − (cid:19) , e ρ ij ( e σ q ) = (cid:18) ζ jq − (cid:19) . ( B . e ρ ij ( e σ p ) = (cid:18) ζ ip − ζ ip − (cid:19) and e ρ ij ( e σ q ) = (cid:18) ζ jq − ζ jq − (cid:19) , we see that the representations e ρ ij with 0 ≤ i < p − and 0 ≤ j < q − , ∤ j are irreducible and are not isomorphic to each other. The number of theserepresentations is | G | . So they are all the irreducible representations of e G ofdimension 2.Similarly, when ( p, q ) = ( − , e ρ of dimension 2 defined by e ρ ( e σ − ) = (cid:18) − (cid:19) , and e ρ ( e σ ) = (cid:18) −
11 0 (cid:19) . ( B . p = − q >
2, all irreducible representations of dimension 2are e ρ j with 0 ≤ j < q − , ∤ j , where e ρ j is defined by e ρ j ( e σ − ) = (cid:18) − (cid:19) and e ρ j ( e σ q ) = (cid:18) ζ jq − (cid:19) . ( B . p = 2, all irreducible representations of dimension 2 are ¯ ρ ij with0 ≤ i ≤ ≤ j < q − , ∤ j , where ¯ ρ ij is defined by¯ ρ ij ( e σ − ) = ( − i I, ¯ ρ ij ( e σ ) = (cid:18) − (cid:19) , ¯ ρ ij ( e σ q ) = (cid:18) ζ jq − (cid:19) . ( B . | e σ p | = 2 | σ p | , all irreducible representations of dimension 2 are ˆ ρ ij with 0 ≤ i < p − , ∤ i and 0 ≤ j < q − , where ˆ ρ ij is defined byˆ ρ ij ( e σ p ) = (cid:18) ζ ip − (cid:19) , ˆ ρ ij ( e σ q ) = (cid:18) ζ jq − − ζ jq − (cid:19) . ( B . Assume v ( p − ≤ v ( q − d = gcd( p − , q − , s = p − d , t = q − d ; us + vt = 1as before. We must have that t is even and u is odd. Let τ = e σ sp · e σ tq and µ = e σ − vp · e σ uq . Then N = h e σ p , e σ q i = h τ , µ i and N ∼ = Z /d Z ⊕ Z / s ( q − Z . Any irreducible representation ρ ij : N −→ C ∗ is of the form ρ ij ( τ ) = ζ id = ζ s ( q − i ( p − q − and ρ ij ( µ ) = ζ j s ( q − = ζ dj ( p − q − . From e σ p = τ u µ − t and e σ q = τ v µ s , we have ρ ij ( e σ p ) = ζ sui − jp − ; ρ ij ( e σ q ) = ζ tvi + j q − ; ρ ij ( ε ) = ρ ij ( e σ p ) p − = ( − j . It is easy to show ρ e σ p ij = ρ ij ⇐⇒ j ≡ . The induced representation e ρ ij : e G −→ GL ( C ) of ρ ij with odd j is given by e ρ ij ( τ ) = (cid:18) ζ id ζ id (cid:19) ; e ρ ij ( µ ) = ζ j s ( q − − ζ j s ( q − ! . Here in the first equality we used the fact that t is even, and in the secondequality we used the fact that u is odd. Furthermore we have e ρ ij ( e σ p ) = (cid:18) ζ sui − jp − (cid:19) ; e ρ ij ( e σ q ) = ζ tvi + j q − − ζ tvi + j q − ! . ( C . e ρ ij at τ and µ , we see that all therepresentations e ρ ij with 0 ≤ i < d and 0 ≤ j < s ( q − , ∤ j are irreducibleand are not isomorphic to each other. The number of these representationsis d · s ( q − = | G | . So they are all the irreducible representations of e G ofdimension 2. 8imilarly, if v ( p − > v ( q − d ′ = gcd( p − , q −
12 ) , s ′ = p − d , t ′ = q − d ; u ′ s ′ + v ′ t ′ = 1 . Then all the irreducible representations of e G of dimension 2 are ˆ ρ ij with0 ≤ i < d ′ and 0 ≤ j < t ′ ( p − , ∤ j , where ˆ ρ ij is defined byˆ ρ ij ( e σ p ) = ζ s ′ u ′ i + j p − − ζ s ′ u ′ i + j p − ! ; ˆ ρ ij ( e σ q ) = (cid:18) ζ t ′ v ′ i − jq − (cid:19) . ( C . ( e G ) be the set of all irreducible representations, up to isomorphism,of e G of dimension 2. As a summary, we have proved the following. Theorem 3.1.
All 2-dimensional irreducible representations of e G are in-duced from the representations of N . In detail, we haveIn the case (A) R ( e G ) = { e ρ j | ≤ j < q − } if p = − { ¯ ρ ij | i = 0 , , ≤ j < q − } if p = 2 { e ρ ij | ≤ i < p − , ≤ j < q − } if p > , where e ρ j , ¯ ρ ij and e ρ ij are defined in (A3.2), (A3.3) and (A3.1) respectively.In the case (B) R ( e G ) = { e ρ } if ( p, q ) = ( − , { e ρ j | ≤ j < q − , ∤ j } if p = − , q > { ¯ ρ ij | i = 0 , , ≤ j < q − , ∤ j } if p = 2 { ˆ ρ ij | ≤ i < p − , ∤ i, ≤ j < q − } if | e σ p | = 2 | σ p |{ e ρ ij | ≤ i < p − , ≤ j < q − , ∤ j } otherwise , where e ρ , e ρ j , ¯ ρ ij , ˆ ρ ij and e ρ ij are defined in (B3.2), (B3.3), (B3.4), (B3.5)and (B3.1) respectively.In the case (C) R ( e G ) = ( { e ρ ij | ≤ i < d, ≤ j < s ( q − , ∤ j } if v ( p − ≤ v ( q − , { ˆ ρ ij | ≤ i < d ′ , ≤ j < t ′ ( p − , ∤ j } otherwise , where e ρ ij and ˆ ρ ij are defined in (C3.1) and (C3.2) respectively. The Frobenious maps
This section is a preparation for the next section to compute the Artin L -functions of the quasi-cyclotomic fields e K when p = −
1. For a prime number ℓ which is unramified in e K/K , let I ℓ (resp. e I ℓ ) be the inertia group of ℓ inthe extension K/ Q (resp. e K/ Q ). Let Fr ℓ be the Frobenious automorphismof ℓ in G/I ℓ and e Fr ℓ the Frobenious automorphism of ℓ in e G/ e I ℓ associated tosome prime ideal over ℓ . In this section we determine e Fr ℓ by Fr ℓ for ℓ = 2.From now on, we always assume that p = −
1, namely, K = Q ( ζ q ) and e K = K ( √ q ∗ ). For a prime number ℓ , we say that ℓ is ramified (resp. inertia,splitting) in the relative quadratic extension ˜ K/K if the prime ideals of K over ℓ are ramified (resp. inertia, splitting) in ˜ K . In [Sect.5, 7] we havedetermined the decomposition nature of odd prime numbers in e K/K . Nowwe determine the decomposition nature of 2 in e K/K . Proposition 4.1. If q = 2 , then 2 is ramified in e K/K . If q is odd, then 2 isunramified in e K/K if and only if ( q ) = 1 , and in this case 2 splits in e K/K if q ∗ ≡ and is inertia in e K/K if q ∗ ≡ but q ∗ .Proof. We first consider the case q = 2. The unique prime ideal of K over 2is the principal ideal generated by π := 1 − ζ . Since the ramification degreeof 2 in K/ Q is 4 and √ π ( π + 2 ζ ) ζ , we have that 2 is ramified in e K/K if and only if x ≡ √ π is not solvable in the ring O K of the integersof K by [2], which is equivalent to that (1 + π ζ ) ζ mod π is not a square.Since 2 = uπ for some unit u , we have(1 + 2 π ζ ) ζ ≡ ζ ≡ (1 − π )mod π , namely (1 + π ζ ) ζ mod π is not a square. So 2 is ramified in e K/K .Now we assume that q is odd. Let π = 1 − ζ . Then 2 is unramified in e K/K if and only if x ≡ √ q ∗ mod π is solvable in O K . Furthermore, 2 splitsin e K/K if and only if x ≡ √ q ∗ mod π is solvable in O K . By Gauss sum wehave √ q ∗ = q − X a =1 ( aq ) ζ aq = 1 + 2 X ( aq )=1 ζ aq . Let α = P ( aq )=1 ζ aq , β = P ( aq )=1 ζ a q , and γ = P ( aq )=1 P ( bq )=1 , a
1. Then α = β − γ ,from which and the equality 2 = π − π , we have √ q ∗ = 1 + 2 β − γ = 1 + π β − π β − γ ≡ (1 + π β ) − π ( β + β ) + π ( β − γ ) ≡ (1 + π β ) − π ( α + β ) + π ( β + γ )mod π . ζ q = − ζ − q − q = − ζ tq , where t is the inverse of 2 in ( Z /q Z ) ∗ , we see β = P ( aq )=1 ( − a ζ taq ≡ P ( aq )=1 ζ taq mod2. So if ( q ) = 1 we have α ≡ β mod2and thus 2 is unramified in e K/K , and if ( q ) = − α + β ≡ P q − a =1 ζ aq = − e K/K .Now we assume ( q ) = 1. Then √ q ∗ mod π is a square if and only if π | β + γ . We consider 2( β + γ ). Since α ≡ β mod2, we have2( β + γ ) = 2 β + β − α ≡ α ( α + 1)mod4 . From √ q ∗ = 1 + 2 α , we see α ( α + 1) = q ∗ − . Since 8 | q ∗ − q ) = 1, we have β + γ ≡ q ∗ − mod2. So π | β + γ if and only if π | q ∗ − , namely 2 | q ∗ − . We complete the proof.By the way, we have determined the ring O e K of the integers of e K . Infact, we have Corollary 4.2.
Assume that q is an odd prime number. Let t q = ( q − / if q ≡ , and t q = ( q − / if q ≡ . Then O e K = Z (cid:20) ζ q , √ q ∗ +1+ π βπ (sin[ q ]) tq (cid:21) if ( q ) = − Z (cid:20) ζ q , √ q ∗ +1+ π β q ]) tq (cid:21) if ( q ) = 1 . Proof.
See [Th.2, 7].Now we assume that 2 is unramified in e K/K . Let Fr ∈ G such thatFr ( ζ ) = 1 and Fr ( ζ q ) = ζ q . It is a Frobenious element of 2 in G modulo I . We have Fr = σ b with 2 | b for ( q ) = 1. Thus e Fr = e σ b or e Fr = e σ b ε .We need to determine e Fr completely. Since ( q ) = 1, we have √ q ∗ ≡ (1 + π α ) + π ( β + γ )mod π . Write u = 1 + π α for simplicity. Since √ q ∗ ≡ u mod π , we see √ q ∗ − u ∈ O e K .Let ℘ be the prime ideal of e K over 2 associated to e Fr . By the definition, wehave e Fr (cid:18) √ q ∗ − u (cid:19) ≡ (cid:18) √ q ∗ − u (cid:19) ≡ ( β + γ ) + √ q ∗ − u ℘. On the other hand, since e σ b q ( √ q ∗ ) = ( − b
22 4 √ q ∗ and e σ b q ( u ) = u for 2 | b ,we have e σ b q ( √ q ∗ − u − b
22 4 √ q ∗ − u e σ b q ε ( √ q ∗ − u − b +1 √ q ∗ − u . So if 2 | b we have e Fr = e σ b if and only if π | β + γ (namely 2 splits in e K/K ), and if 2 ∤ b we have e Fr = e σ b if and only if π ∤ β + γ (namely 2 isinertia in e K/K ). In the case q ≡ ∤ b ,since if 4 | b , we may replace b by b + ( q − q ≡ | b ⇐⇒ q − ≡ q ⇐⇒ q has the form A + 64 B for A, B ∈ Z ,by the Exercise 28 in Chap.5 in [5]. So we get the following result Proposition 4.3.
Assume that 2 is unramified in e K/K . Let Fr = σ b . Wehave | b . If q ≡ , we always assume b ≡ . Let P be the setof the prime numbers of the form A + 64 B with A, B ∈ Z . Then we have e Fr = (e σ b if q P , ∤ q ∗ − , or q ∈ P , | q ∗ − e σ b ε if q ∈ P , ∤ q ∗ − , or q P , | q ∗ − . The following lemma is useful in next section.
Lemma 4.4.
We have ε ∈ e I ℓ if and only if ℓ is ramified in e K/K .Proof.
The canonical projection e G −→ G ≃ e G/ h ε i induces a surjective homo-morphism e I ℓ −→ I ℓ which implies the isomorphism e I ℓ / < ε > ∩ e I ℓ ∼ = I ℓ . Thus ℓ is ramified in e K/K ⇐⇒ | e I ℓ | = 2 | I ℓ | ⇐⇒ | e I ℓ ∩ < ε > | = 2 ⇐⇒ ε ∈ e I ℓ . L -functions In this section we compute the Artin L -functions of the quasi-cyclotomicfields e K = Q ( ζ q , √ q ∗ ).The L -functions associated to the 1-dimensional representations of e G arethe well-known Dirichlet L -functions. So we mainly compute the L -functionsassociated to the 2-dimensional irreducible representation of e G . Let ϕ : e G → GL( V ) be a 2-dimensional irreducible representations. The Artin L -function L ( ϕ, s ) associated to ϕ is defined as the product of the local factors L ( ϕ, s ) = Y ℓ :prime L ℓ ( ϕ, s ) , where the local factors are defined as L ℓ ( ϕ, s ) = det(1 − ϕ ( e Fr ℓ ) ℓ − s | V e I ℓ ) − .Now we begin to compute them. First we notice that if ℓ is ramified in e K/K , then V e I ℓ = 0 and L ℓ ( ϕ, s ) = 1, which is due to the facts that ε ∈ e I ℓ byLem.4.4 and ϕ ( ε ) = − I for any irreducible representation ϕ of e G by Th.3.1.12 .1 The case q = 2 . By section 3, there is only one 2-dimensional represen-tation e ρ in this case, which is defined as e ρ ( e σ − ) = (cid:18) − (cid:19) , and e ρ ( e σ ) = (cid:18) −
11 0 (cid:19) . Since 2 is ramified in e K/K , we have L ( e ρ , s ) = 1. Assume that ℓ is an oddprime number.If ℓ ≡ ℓ = σ − and thus e Fr ℓ = e σ − or e σ − ε . In any casewe have L ℓ ( e ρ , s ) = det (cid:18) I ± (cid:18) − (cid:19) ℓ − s (cid:19) − = (1 − ℓ − s ) − . If ℓ ≡ ℓ = σ and thus e Fr ℓ = e σ or e σ ε . We have L ℓ ( e ρ , s ) = det (cid:18) I ± (cid:18) −
11 0 (cid:19) ℓ − s (cid:19) − = (1 + ℓ − s ) − . If ℓ ≡ ℓ = σ − σ and thus e Fr ℓ = e σ − e σ or e σ − e σ ε . Wehave L ℓ ( e ρ , s ) = det (cid:18) I ± (cid:18) − (cid:19) (cid:18) −
11 0 (cid:19) ℓ − s (cid:19) − = (1 − ℓ − s ) − . If ℓ ≡ ℓ = 1 and thus e Fr ℓ = 1 or ε . In this case, wemust determine e Fr ℓ completely. Since e Fr ℓ ( √ ≡ ( √ ℓ mod ℘ for the primeideal ℘ of e K over ℓ associated to e Fr ℓ , we have e Fr ℓ = 1 if 2 ℓ − ≡ ℓ , and e Fr ℓ = ε if 2 ℓ − ≡ − ℓ . As in last section, we have that for ℓ ≡ ℓ − ≡ ℓ if and only if ℓ ∈ P . So we have L ℓ ( e ρ , s ) = ( (1 − ℓ − s ) − if ℓ ∈ P (1 + ℓ − s ) − otherwise.We get the Artin L -function in the case ( p, q ) = ( − ,
2) as follows. L ( e ρ , s ) = Y ℓ ≡ (1 − ℓ − s ) − · Y ℓ ≡ (1 + ℓ − s ) − × Y ℓ ∈ P (1 − ℓ − s ) − · Y ℓ ≡ , ℓ P (1 + ℓ − s ) − . (5.1) q is odd. All 2-dimensional irreducible representations of e G are e ρ j with 0 ≤ j < q − , | j if q ≡ ≤ j < q − , ∤ j if q ≡ e ρ j is defined by e ρ j ( e σ − ) = (cid:18) − (cid:19) , e ρ j ( e σ q ) = (cid:18) ζ jq − (cid:19) and e ρ j ( ε ) = − I.
13e first determine the local factors L ℓ ( e ρ j , s ) for ℓ = 2 , q . For such ℓ , wehave V e I ℓ = V . Let Fr ℓ = σ a ℓ − σ b ℓ q , which is equivalent to ℓ ≡ ( − a ℓ mod4 and ℓ ≡ g b ℓ mod q , where g is the primitive root mod q associated to σ q . It is easyto compute that e ρ j ( e σ b ℓ q ) = (cid:18) ζ jq − (cid:19) b ℓ = ζ jb ℓ q − I if 2 | b ℓ ζ j ( b ℓ +1)2( q − ζ j ( b ℓ − q − ! if 2 ∤ b ℓ . Furthermore, we havedet( I − e ρ j ( e σ a ℓ − e σ b ℓ q ) ℓ − s ) = (1 − ζ jb ℓ q − ℓ − s ) if a ℓ = 0 , | b ℓ − ζ jb ℓ q − ℓ − s if a ℓ = 0 , ∤ b ℓ or a ℓ = 1 , | b ℓ ζ jb ℓ q − ℓ − s if a ℓ = 1 , ∤ b ℓ and det( I + e ρ j ( e σ a ℓ − e σ b ℓ q ) ℓ − s ) = (1 + ζ jb ℓ q − ℓ − s ) if a ℓ = 0 , | b ℓ − ζ jb ℓ q − ℓ − s if a ℓ = 0 , ∤ b ℓ or a ℓ = 1 , | b ℓ ζ jb ℓ q − ℓ − s if a ℓ = 1 , ∤ b ℓ . So we get L ℓ ( e ρ j , s ) = (1 − ζ jb ℓ q − ℓ − s ) − if ℓ ≡ ℓ ≡ g b ℓ mod q with 2 ∤ b ℓ , or if ℓ ≡ ℓ ≡ g b ℓ mod q with 2 | b ℓ , and L ℓ ( e ρ j , s ) = (1 + ζ jb ℓ q − ℓ − s ) − if ℓ ≡ ℓ ≡ g b ℓ mod q with 2 ∤ b ℓ .To compute the local factors when ℓ ≡ ℓ ≡ g b ℓ mod q with2 | b ℓ , we must determine e Fr ℓ completely. Since ( ℓq ) = 1, we have ( qℓ ) = 1 and( q ∗ ℓ ) = 1. Let α ℓ ∈ Z such that α ℓ ≡ q ∗ mod ℓ . From e σ b ℓ q ( √ q ∗ ) = ( − bℓ √ q ∗ ,we see e Fr ℓ = e σ b ℓ q if ( α ℓ ℓ ) = ( − bℓ , and e Fr ℓ = e σ b ℓ q ε if ( α ℓ ℓ ) = ( − bℓ +1 . Sowhen ℓ ≡ ℓ ≡ g b ℓ mod q with 2 | b ℓ , we have L ℓ ( e ρ j , s ) = ( (1 − ζ jb ℓ q − ℓ − s ) − if ( α ℓ ℓ ) = ( − bℓ (1 + ζ jb ℓ q − ℓ − s ) − if ( α ℓ ℓ ) = ( − bℓ +1 . Next we compute the local factors L ( e ρ j , s ) and L q ( e ρ j , s ). When ( q ) = −
1, we know from last section that 2 is ramified in e K/K . So L ( e ρ j , s ) = 114n this case. Now we assume ( q ) = 1. Since I = < σ − > and 2 is unramifedin e K/K , we have e I = < e σ − > or e I = < e σ − ε > . The matrixes I + e ρ j ( e σ − )and I + e ρ j ( e σ − ε ) have rank 1. So V e I has dimension 1. Write Fr = σ b with 2 | b . As in last section, we always assume b ≡ q ≡ P be the set of the prime numbers of the form A + 64 B with A, B ∈ Z . Since e ρ j ( e σ b ) = ζ jb q − I , by Prop.4.3 we have L ( e ρ j , s ) = ( − ζ jb q − − s if q P , ∤ q ∗ − , or q ∈ P , | q ∗ −
11 + ζ jb q − − s if q ∈ P , ∤ q ∗ − , or q P , | q ∗ − . When q ≡ q is ramified in e K/K . So L q ( e ρ j , s ) = 1for odd j in this case. Assume q ≡ I q = < σ q > and q isunramifed in e K/K , we have e I q = < e σ q > or e I = < e σ q ε > . Thus V e I q = 0 if j = 0, and V e I q has dimension 1 if j = 0.The Frobenious map Fr q of q in G modulo I q is the identity map. So e Fr q = 1 or ε . In [Sect.5, 7] we have showed that q splits in e K/K if q ≡ q ≡ e Fr = 1 if q ≡ e Fr = ε if q ≡ L q ( e ρ j , s ) = j = 01 − q − s if j = 0 , q ≡ q − s if j = 0 , q ≡ . We have computed all the local factors. So we have L ( e ρ j , s ) =(1 − u q ζ jb q − − s ) − (1 − ( − q − q − s ) − n j × Y ℓ ≡ , ∤ b ℓ or ℓ ≡ , | b ℓ (1 − ζ jb ℓ q − ℓ − s ) − × Y ℓ ≡ , ∤ b ℓ (1 + ζ jb ℓ q − ℓ − s ) − Y ℓ ≡ , | b ℓ (1 − u ℓ ζ jb ℓ q − ℓ − s ) − , (5.2)where u q = 1 if q P , ∤ q ∗ − , or q ∈ P , | q ∗ − u q = − n j = 0 if j = 0 and n = 1; and u ℓ = ( α ℓ ℓ )( − bℓ . Here in theproducts, ” ≡ ” means the congruence modulo 4. Theorem 5.1.
Except for the Dirichlet L -functions, all Artin L -functions ofthe Galois extension e K/ Q are explicitly given by (5.1) in the case q = 2 andby (5.2) in the case q is odd, where in (5.2) ≤ j < q − , | j if q ≡ and ≤ j < q − , ∤ j if q ≡ . .3 A formula. Let ζ e K ( s ) and ζ K ( s ) be the Dedekind zeta functions of e K and K respectively. By Artin’s formula of the decomposition of Dedekindzeta functions, we have ζ e K ( s ) ζ K ( s ) = Y e ρ j Y ℓ : prime L ℓ ( e ρ j , s ) , where e ρ j runs over all 2-dimensional irreducible representations of e G . When q = 2, there is only one 2-dimensional irreducible representation of e G . So thesquare of (5.1) gives the formula. When q is odd, by computing Q e ρ j L ℓ ( e ρ j , s ),we get the following Corollary 5.2.
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