The Maximal Discrete Extension of the Hermitian Modular Group

Home , Modular group

arXiv:1910.12466v2 [math.NT] 27 Jan 2021 Abstract. mgnr-udai ubrfield number imaginary-quadratic ahmtc ujc Classification. Subject Mathematics group orthogonal involution, Lehner Keywords. h case the xeso in extension vrapriua ubrfield number particular a over nouin.Mroe efidantrlcaatrzto of characterization natural a find we Moreover involutions. inivle the involves tion 1 ly re,Anln Wernz, Annalena Krieg, Aloys naeaWrzwsprilyspotdb Graduiertenkoll by supported partially Vetenskapsr˚adet was by Wernz Gr supported Annalena partially was Raum Martin atnRaum, Martin h aia iceeetnino the of extension discrete maximal The er tRT ahnUniversity. Aachen RWTH at gebra [email protected] Göteborg, Sweden 96, SE-412 hlestkik ¨gkl c ¨tbrsUiestt I Göteborgs Universitet, högskola och tekniska Chalmers D-520 [email protected] University, [email protected], Aachen RWTH für Mathematik, A Lehrstuhl n egv nepii ecito,wihivle generali involves which description, explicit an give we 2 = ly Krieg Aloys SU e Γ Let emta oua ru,nraie,mxmldsrt ext discrete maximal normalizer, group, modular Hermitian emta oua group modular Hermitian ( n ,n n, trinsbru fteielcasgopof group class ideal the of subgroup -torsion n ( ; O C K ,wihcicdswt h omlzro Γ of normalizer the with coincides which ), eoeteHriinmdlrgopo degree of group modular Hermitian the denote ) , atnRaum Martin K b n K n ecndsrb h aie rmsi t In it. in primes ramified the describe can we and nti ae edtrieismxmldiscrete maximal its determine we paper this In . by 10,11F55 11F06, 1 and siuinnfo aeaik vetenskaper, för Matematiska nstitutionen naeaWernz Annalena gEprmnel n osrkieAl- konstruktive und Experimentelle eg n 2015-04139. ant 6Ace,Germany Aachen, 56 hsgopin group this K hsgopi defined is group This . n ( O e Atkin-Lehner zed K 1 .Tedescrip- The ). nin Atkin- ension, SO (2 n , vran over 4). The Hermitian modular group of degree n over an imaginary-quadratic number field K was introduced by H. Braun [2]. We determine its maximal discrete extension in SU(n,n; C), which is called the extended Hermitian modular group. In the proof we follow ideas presented in [5]. The extended Hermitian modular group coincides with the normalizer of the Hermitian modular group in SU(n,n; C). Its entries belong to the ring of integers of an interesting number field Kn depending on K and n. We can show that the field extension K Q is unramified outside ndK. n ⊇ Due to the work of Borcherds there is specialb interest in the case n = 2, which can be viewed as an orthogonalb group SO(2, 4). In this paper we derive an explicit isomorphism. We describe the extended Hermitian modular group of degree 2 explicitly by means of generalized Atkin-Lehner involutions and show that it has got a natural description in the orthogonal context.

The special unitary group SU(n,n; C) consists of all matrices

1 0 tr (1) M = A B SL (C) satisfying M JM = J, J = 0 I , I = .. , C D ∈ 2n I −0 . 0 1 where the blocks A,B,C,D are always square matrices.

Lemma 1. Given M = A B SU(n,n; C) then C D ∈ det A, det B, det C, det D R. ∈ Proof. It suﬃces to show the result for the A-block, because the other cases are obtained from multiplication with J. As det A = 0 is clear, let det A = 0. Then we have 6 1 I 0 I A− B A 0 1 M − = tr 1 CA− I 0 I 0 A − − ! due to (1). Hence 1 = det M = (det A) (det A) yields the claim. The group SU(n,n; C) acts on the Hermitian half-space (cf. [2])

n n 1 tr H := Z C × ; (Z Z ) positive deﬁnite n { ∈ 2i − } via 1 Z M Z = (AZ + B)(CZ + D)− . 7→ h i Throughout this paper let

K = Q(√ m) C, m N squarefree, − ⊂ ∈

2 be an imaginary-quadratic number ﬁeld. Its discriminant and ring of integers are

m Z + Z(1 + √ m)/2, if m 3 (mod 4), dK = − and K = Z + ZωK = − ≡ 4m O Z + Z√ m, if m 1, 2 (mod 4). (− ( − ≡ Denote its unit group by . OK∗ The Hermitian modular group of degree n is given by

2n 2n Γ ( K) := SU(n,n; C) × . n O ∩ OK

It is well-known that Γ ( K)= SL (Z). 1 O 2

Lemma 2. Let ∆ K be a discrete subgroup of SU(n,n; C) containing Γ ( K) or a n, n O subgroup of SU(n,n; C), which contains Γ ( K) as a normal subgroup. Given M ∆ K n O ∈ n, there exists u C 0 such that ∈ \{ } 2n 2n uM × . ∈ OK Any such u satisﬁes ℓ = u 2 N. | | ∈ Proof. Γ ( K) possesses a fundamental domain of ﬁnite positive volume in H due to n O n [2]. If ∆n,K is discrete, we can proceed in exactly the same way as Ramanathan [20] in the proof of his Theorem 1. We have r = [∆ K :Γ ( K)] < and conclude n, n O ∞ 1 s (MRM − ) Γ ( K) for all R Γ ( K), s := r! . ∈ n O ∈ n O We use this for

IH I 0 tr n n (2) R = , ,H = H × . 0 I H I ∈ OK A∗ B∗ If R = C∗ D∗ , A∗ = (aij), we end up with sa a K for all i,j,k,ℓ = 1,...,n. ij kℓ ∈ O The same holds for B , C , D . Thus the existence of u C 0 satisfying ∗ ∗ ∗ ∈ \{ } 2n 2n uM = L × ∈ OK follows. The identity tr u 2J = (uM)trJ(uM)= L JL | | 2 yields u K R = Z, because the elements on the right hand side are integral. If | | ∈ O ∩ Γ ( K) is normal in ∆ K, we can take the same arguments with s = 1. n O n, We want to simplify the shape of M. We cite a result of Ensenbach [6] (cf. [22]) as

3 Lemma 3. Given M ∆ K there exists an R Γ ( K) such that ∈ n, ∈ n O

A∗ B∗ RM = , A∗ SLn(C). 0 D∗ ∈ A B 2n 2n tr tr Proof. Apply Lemma 2 and let L = uM = × . Then (C , A ) is a C D ∈ OK Hermitian pair in the sense of Braun [1]. Due to [1], Theorem 3, there exists a coprime pair in this class, which can be completed to a matrix in Γ ( K). Multiplying by its n O inverse yields the shape of RM. If∆n,K is discrete, we obtain det A∗ = 1 as the index | | 2 [∆ K :Γ ( K)] is ﬁnite. If Γ ( K) is nomal in ∆ K then (2) for H = I yields det A , n, n O n O n, | ∗| det D 2 Z, hence det A = 1. In view of Lemma 1 we may multiply by | ∗| ∈ | ∗| P 0 , P = diag (1,..., 1, 1), 0 P − in order to obtain det A∗ = 1. We have a closer look at a special case.

Lemma 4. Let ∆ K be a discrete subgroup of SU(n,n; K) containing Γ ( K). Then n, n O

∆ K =Γ ( K). n, n O

Proof. We assume that there exists an M ∆ K with an entry in K K. Then there ∈ n, \O is a prime ideal ℘ K and an entry x of M = (m ) such that the exponent e(x,℘) O ij ⊆ A B of the prime ideal decomposition of Kx is < 0. Choose M = , A = (a ), as in O 0 D ij Lemma 3 with C = 0 and assume without restriction that e(a ,℘) 6 min e(m ,℘), 1 6 i, j 6 2n . 11 { ij } After multiplying with matrices

tr tr U 0 1 0 1 g n 1 ,U = , , g − , 0 U 1 g I 0 I ∈ OK − we may assume

e(a ,℘) < min e(a ,℘), e(a ,℘), j = 2,...,n . 11 { j1 1j } Considering Ar = (a ), r N, we get ij∗ ∈

e(a∗ ,℘)= r e(a ,℘) 6 e(a∗ ,℘), 1 6 i, j 6 n , 11 11 ij } e(a∗ ,℘) < min e(a∗ ,℘), e(a∗ ,℘), j = 2,...,n . 11 { j1 1j } Since the multiplication by unimodular matrices does not change the ideal generated by the entries of M r, the cosets r Γ ( K)M , r N , n O ∈ 0

4 are mutually distinct. This contradicts

[∆ K :Γ ( K)] < . n, n O ∞ An immediate application is

Corollary 1. Let ∆ K be a discrete subgroup of SU(n,n; C) containing Γ ( K). Then n, n O ∆ K is contained in the normalizer of Γ ( K) in SU(n,n; C). n, n O 1 2n 2n Proof. Let M ∆ K, u C 0 , M = L, L × according to Lemma 2. Then ∈ n, ∈ \{ } u ∈ OK 1 1 MΓ ( K)M − = LΓ ( K)L− ∆ K SU(n,n; K). n O n O ⊆ n, ∩ 1 Hence we can apply Lemma 4 to the group generated by Γ ( K) and MΓ ( K)M . n O n O − The result is 1 MΓ ( K)M − =Γ ( K). n O n O 1 2n 2n Now let M = L, L × , belong to the normalizer of Γ ( K) in SU(n,n; C), u ∈ OK n O ℓ = uu N and let (L) stand for the ideal generated by the entries of L. If denotes ∈ I N the reduced norm of an ideal in K, we obtain O

Lemma 5. (L) is an invariant of the double coset with respect to Γ ( K). One has I n O a) ℓ = ( (L)), ℓ K = (L) (L), O n N I n I n · I b) u K, u K = (L) . ∈ O O I Proof. (L) does not change, if we multiply by unimodular matrices. Using Lemma 3 I 1 with A∗ = u A we get n u = det A K. ∈ O tr Then AD = ℓI yields ℓ (L) (L). As the proof of Lemma 2 shows ∈I · I

xy ℓ K for all entries x,y of L, ∈ O we obtain (L) (L) ℓ K I · I ⊆ O and therefore equality. Computing the reduced norm we get N 2 2 ℓ = (ℓ K)= ( (L) (L)) = ( (L)) . N O N I · I N I Computing the determinant we obtain

n n n n u = det A (L) , i.e. u K (L) ∈I O ⊆I hence n 2 n n n ℓ = det A = (u K) > ( (L)) = ℓ . | | N O N I n n Therefore u K = (L) follows. O I

5 Let ℓK = [ ]; fractional ideal in K C { A A } stand for the ideal class group of K with class number hK. Let

n ℓK[n] := [ ] ℓK; [ ] = [ K] C { A ∈C A O } be the subgroup of elements with order dividing n, the n-torsion subgroup. Moreover stands for a cyclic group of order r. Cr Theorem 1. Let K be an imaginary-quadratic number ﬁeld. a) The extension

1 2n 2n n n n ∆∗ := L SU(n,n; C); L × , u C, u K, u K = (L) n,K u ∈ ∈ OK ∈ ∈ O O I is a subgroup of SU(n,n; C) containing Γ ( K). n O b) The map 1 ∆∗ ℓK[n], M = L [ (L)], n,K →C u 7→ I is a surjective homomophism of the groups. If dK = 3, 4 its kernel is equal to 6 − − 2n εM; ε C, ε = 1, M Γ ( K) . { ∈ ∈ n O } c) ∆ is the maximal discrete extension of Γ ( K) in SU(n,n; C) and coincides with n,∗ K n O the normalizer of Γ ( K) in SU(n,n; C). The factor group ∆ /Γ ( K) is isomorphic n O n,∗ K n O to ℓK[n], if dK = 3, 4. Cn ×C 6 − −

We call ∆n,∗ K the extended Hermitian modular group of degree n.

1 1 Proof. a) Given M = L, M = ′ L ∆ we have u ′ u ′ ∈ n,∗ K n n n n n n (uu′) K (LL′) (L) (L′) = u K u′ K, O ⊆I ⊆I ·I O · O hence MM ∆ . The result follows from ′ ∈ n,∗ K tr tr 1 D B M − = tr − tr . C A − ! 1 1 b) At ﬁrst M = u L = u′ L′ with integral L,L′ implies

u′ 1 I = L′L− GL (K), u ∈ 2n hence u′ u′ K and K (L)= (L′). u ∈ u O ·I I Thus [ (L)] ℓK[n] is well-deﬁned by M. Hence the map is a homomophism of the I ∈ C

6 groups due to Lemma 5 and a). If dK = 3, 4 we have 6 − − n n n n 2n u K = v K u = v u = εv, ε = 1. O O ⇔ ± ⇔

Given [ ] K[n] we may assume K. According to [7], Satz 2.1, resp. [21] there A ∈I n n A ⊆ O n is a matrix A × with elementary divisors ,..., and K det A = , i.e. ∈ OK A A O A n n n n (3) A × , K det A = , u := √det A C, ℓ = uu = ( ). ∈A O A ∈ N A 1 n n n n 1 tr Then we have AGA × for all G × as well as AA SL ( K). Thus − ∈ OK ∈ OK ℓ ∈ n O

1 A 0 tr 1 M = ∆∗ , D = ℓA − u 0 D ∈ n,K follows. As this M is mapped onto [ ], the map is surjective. A c) Observe that the image consists of at most hK elements. Thus

[∆∗ :Γ ( K)] < n,K n O ∞ follows. Hence ∆ is discrete. Thus it coincides with the normalizer of Γ ( K) accord- n,∗ K n O ing to Corollary 1. The result on the factor group follows from b).

It is a consequence of the proof that representatives of the Γ ( K)-cosets in ∆ can n O n,∗ K be given in the form

tr A 0 1 n n n (4) , A = L SL (C), u C, L × , (L) = K det L. 0 A 1 u ∈ n ∈ ∈ OK I O −

We formulate a special case and include the cases dK = 3, 4 from Theorem 1 in − − Corollary 2. Let K be an imaginary-quadratic number ﬁeld such that the class number hK is coprime to n. Then the extended Hermitian modular group is equal to

2n 2 diag (ε, . . . , ε, εδ, ε, . . . , ε, εδ)M; M Γ ( K), ε C, δ ∗ , ε δ = 1 . ∈ n O ∈ ∈ OK If n = 1 our result reproves the fact that SL (Z)=Γ ( K)=∆ coincides with its 2 1 O 1∗,K normalizer and maximal discrete extension in SL2(R). The same arguments as in (4) can be used in order to obtain

Corollary 3. Let K be an imaginary-quadratic number ﬁeld. Then the maximal discrete extension of SL ( K) in SL (C) coincides with the normalizer of SL ( K) in SL (C) n O n n O n and is given by

1 n n n L SL (C); L × , 0 = u C, K det L = (L) . u ∈ n ∈ OK 6 ∈ O I The explicit description of this group in the case n = 2 is contained in [19].

7 Remark 1. a) The result for SU(n,n; K) diﬀers from the corresponding result for Sp(n; K) in [20], Theorem 2, as the index [∆ : Γ ( K)] is not independent of n. n,∗ K n O Moreover the factor group does not only contain elements of order 1 or 2, if n> 2. b) The entries of the coset tr A 0 Γ ( K) 0 A 1 n O − in (4) are algebraic integers of the algebraic number ﬁeld Q(√ m, u) of degree 6 2n, n − because (a/u) K for any entry a of A. ∈ O c) It follows from (4) that

n M Γ ( K) for all M ∆∗ . ∈ n O ∈ n,K Now we have a closer look at the Hermitian modular group for n = 2. Let d N be a ∈ squarefree divisor of dK. Then

= Kd + K(m + √ m) K Ad O O − ⊆ O is the unique ideal of reduced norm d in K. If e dK is squarefree and coprime to d the O | prime ideal decomposition yields

2 = d K, = . Ad O Ad ·Ae Ade Following [19] we determine u, v Z such that ∈ ud v(m2 + m)/d = 1, − i.e.

tr 1 ud v(m+√ m) V d 0 (5) V := − SL (C), W := −1 SU(2, 2; C). d m √ m d 2 d 0 V √d − − ∈ d ∈ Then we get

1 4 4 Γ ( K)W = W Γ ( K)= M SU(2, 2; C); M × . 2 O d d 2 O √ ∈ ∈Ad d Theorem 2. For an imaginary-quadratic number ﬁeld K, the extended Hermitian modular group of degree 2 is given by

∆∗ = Γ ( K)W . 2,K 2 O d d dK,d free | [− ν It contains Γ ( K) as a normal subgroup of index 2 , 2 O

ν := ♯ p; p prime, p dK , { | }

8 and the factor group satisﬁes ν ∆∗ /Γ ( K) = . 2,K 2 O ∼ C2 Proof. The result follows from Theorem 1, Corollary 2 and [19]. But we also give a direct proof. Let 1 1 A∗ B∗ M = u L = u ∆2∗,K 0 D∗ ∈ according to Theorem 1 and Lemma 3. Then (A )= ℓ = uu N and det A 2 = ℓ2. N ∗ ∈ | ∗| If we consider (ℓ + det A∗)M instead of M, if necessary, we may assume det A∗ N. 1 2 2 ∈ Cancelling integers we may even assume A × for r> 1, hence r ∗ 6∈ OK 1 M = L. √ℓ

2 In view of ℓ K = (L) due to Theorem 1, we conclude that ℓ is a squarefree divisor of O I dK from the prime ideal decomposition in K. O We consider particular examples in

Remark 2. a) One has iI Γ ( K)W , ∈ 2 O m hence ∆∗ =Γ ( K) (iI)Γ ( K), 2,K 2 O ∪ 2 O if dK = 4 is a prime discriminant. If dK = 4 one has 6 − −

1+i U 0 1 0 Γ ( K)W ,U = , √2 0 U ∈ 2 O 2 0 i and therefore 1+i U 0 ∆∗ =Γ ( K) Γ ( K). 2,K 2 O ∪ √2 0 U 2 O This group is isomorphic to U(2, 2; K) and already appeared, when the attached graded O ring of Hermitian modular forms was determined (cf. [9]). b) Theorem 1 and Theorem 2 reprove a result of Hecke that

ν 1 ℓK[2] = − . C ∼ C2 c) If ℓK = ℓK[2] we conclude that C C

εI Γ ( K), if n is odd, 2 O ε C,ε2n=1 · ∆ =  ∈ n,∗ K S εI W Γ ( K), if n is even,  d n O  ε C,ε2n=1 ·  ∈ d dK, dS free | − c  

9 where tr tr 1 1 W = diag (V ,..., V ,V − ,...,V − ) SU(n,n; C). d d d d d ∈ Moreover therec is another group involved in this context, namely

1 4 4 K := M SU(2, 2; C); k N, M × SU(2, 2; K), G √ ∈ ∈ ∈ OK ⊇ k which naturally appears in the associated Hecke theory (cf. [6], [22]).

Now we consider the orthogonal setting. Let

0 0 1 0 0 1 2 2Re(ωK) S = , S0 = 0 S 0 , S1 = 0 S0 0 . 2Re(ωK) 2 ωKωK  −    1 0 0 1 0 0     Then S1 is of signature (2, 4). We deﬁne the attached special orthogonal group by

SO(S ; R) := M SL (R); S [M] := M trS M = S 1 { ∈ 6 1 1 1 1} and denote by SO0(S1; R) the connected component of the identity matrix I. Let

6 6 ΣK := M SO (S ; R); M I + Z × S { ∈ 0 1 ∈ 1} stand for discriminant kernel and ΣK∗ := SO0(S1; Z) for the full group of integral matrices in SO0(S1; R). The group SO0(S1; R) acts on the orthogonal half-space (cf. [12])

tr 4 tr K := z = x + iy = (τ ,z,w,τ ) C , Im τ > 0, y S y > 0 H { 1 2 ∈ 1 0 } via 1 1 z M z := S0[z]b + Kz + c , 7→ h i M z − 2 { } where f tr α a S0 β f γ tr M = b K c , M z = 2 S0[z]+ d S0z + δ.  tr  { } − γ d S0 δ f   f We consider the bijection between the (complexiﬁed) Hermitian 2 2 matrices and R4 × (resp. C4) α β + γω ϕ : K (α,β,γ,δ)tr , β + γωK δ 7→ which satisﬁes ϕ(H )= K. 2 H There is an isomorphism between SU(2, 2; C)/ I and SO (S ; R), where M M, {± } 0 1 ± 7→ f

10 given by

(6) ϕ(M Z )= M ϕ(Z) , det(CZ + D)= M ϕ(Z) for all Z H . h i h i { } ∈ 2 Just as in [10] we obtainf an explicit version, if we usef the abbreviation

♯ α β δ β = − γ δ γ α − for the adjoint matrix, via φ(Z)= z

det(CZ + D)= M z , (AZ + B)(CZ + D)♯ = 1 S [z]b + Kz + c, { } − 2 0 where f α = det A, β = det B, γ = det C, δ = det D, − − (7) a = ϕ(A♯B), b = ϕ(AC♯), c = ϕ(BD♯), d = ϕ(C♯D),  − − ♯ ♯ ♯  Kz = ϕ(AZD + BZ C ). Theorem 3. Let K be an imaginary-quadratic number ﬁeld. Then the map

φ : SU(2, 2; C) SO (S ; R), M M, → 0 1 7→ given by (6) and (7) is a surjective homomorphism of the groupsf with kernel I . It {± } satisﬁes

φ( K)= SO (S ; Q), G 0 1 φ(∆2∗,K)=ΣK∗ ,

φ(Γ ( K))=ΣK. 2 O Proof. The groups on the left hand side are generated by the matrices

tr IH tr U 0 J, ,H = H , , det U R 0 , 0 I 0 U 1 ∈ \{ } − with coeﬃcients in the appropriate set. This is clear for SU(2, 2; C) (cf. [17]), with a slight adaption also for K, for Γ ( K) due to [14], Theorem 3, and for ∆ by virtue G 2 O 2∗,K of Theorem 2. Hence the image is contained in the right hand side in any case. On the other hand it follows from [18] that the groups on the right hand side are generated by matrices of the type

0 0 P 1 λtrS 1 S [λ] 1 0 0 0 1 0 − 2 0 J = 0 I 0 , P = − , 0 I λ , 0 K 0 ,   1 0     P 0 0 − 0 0 1 0 0 1 e       K SO (1, 3). ∈ 0

11 They appear as images of elements in the groups on the left hand side due to (7), if one invokes [19] for the last type of matrices. As the kernel of φ is clearly I , the claim {± } follows.

Remark 3. a) By Theorem 1 and 3 clearly ΣK∗ is the normalizer and the maximal discrete extension of ΣK in SO0(S1; R). b) The result on the normalizer for n = 2 is contained in [24] with a completely different proof. Applications to Hermitian modular forms are described in [25]. It is clear from (4) that the Siegel-Eisenstein series for Γ ( K) (cf. [1]) is a modular form with respect n O to the extended Hermitian modular group ∆n,∗ K. c) Köhler [16] showed that the paramodular group of degree 2 and squarefree level N can be embedded into the Hermitian modular group Γ ( K), if N = uu for some u K. 2 O ∈ O The normalizer of the paramodular group (cf. [15]) can be embedded into ∆2∗,K, when- ever m = N. d) Theorem 3 illustrates that the exceptional isogeny between the real Lie groups SU(2, 2; C) and SO0(S1; R) in general does not decend to an isogeny of algebraic groups (cf. [11]). Indeed, when constructing the underlying isomorphism of the special unitary and the spin group, one typically employs a diagonal quadratic from of signature (2, 4) as opposed to the one that is associated with S1. There is no analogue of the isomorphism between SU(2, 2; C)/ I and SO (S ; R) {± } 0 1 for general n, which we utilized when inspecting the special case n = 2. Let µ ⊆ C be the set of 2n-th roots of unity. Theorem 1 connects the quotient group ∆n,∗ K/ µΓ ( K) with the n-torsion subgroup of the ideal class group of K if dK = 3, 4. n O 6 − − Class groups of imaginary-quadratic number fields including their n-torsion are elusive, only conjecturally governed by the Cohen-Lehnstra heuristic [4] and its recent global refinement [13]. For this reason a detailed description of ∆n,∗ K for general n is out of reach. We next narrow down the field of definition of ∆n,∗ K and provide an algorithm to compute representatives of ∆ /µΓ ( K) for given K and n. n,∗ K n O The field of definition of ∆n,∗ K is

(8) K = K ∆∗ := Q m ; M = (m ) ∆∗ , 1 ≦ i, j 6 2n C. n n,K i,j ij ∈ n,K ⊆ Clearly Remarkb b 1 b) yields 2n 2n ∆n,∗ K b × . ⊆ OKn

Obviously K Kn holds for n > 2. We quote some examples from our considerations ⊆ 2πi/r above. Therefore let ζr = e stand for a primitive r-th root of unity. b Examples. a) Clearly K1 = Q holds. b) If n = 2 Theorem 2 yields b K = Q(ζ , √p; p prime, p dK). 2 4 | b

12 c) One has for n > 2

K[ζ4n], if dK = 4, Kn = − K[ζ ], if dK = 3. ( 6n − If hK is coprime to n and n >b2, dK = 3, 4, then 6 − −

Kn = K[ζ2n] holds. b d) If ℓK = ℓK[2] and n > 2, dK = 3, 4, we have C C 6 − −

K[ζ2n], if n is odd, Kn = (K[ζ2n, √p, p prime, p dK], if n is even. | b The examples show that Kn is an interesting number field. We want to generalize the fact that only ramified primes appear in the definition of K2 to the case of arbitrary n. b Theorem 4. Let K be an imaginary-quadratic number field.b Then the field extension

K = K(∆∗ ) Q n n,K ⊇ is unramified outside of ndK. b b Proof. Recall that the composite of two field extension F K and F K is unramified ⊇ ′ ⊇ at a prime ideal p if and only if both F and F are so. ⊆ OK ′ Let n > 2, as K = Q is trivial. Moreover dK = 3 and dK = 4 are clear due to the 1 − − Example. Therefore let dK = 3, 4. We have K K(∆ ) and the matrix entries of 6 − − ⊆ n,∗ K elements of Γ ( bK) lie in K. Observe that K is unramified outside of dK. The cyclotomic n O field Q(µ) is unramified outside of ndK. Therefore, we can focus on the ramification of fields generated by representatives of ∆ /µΓ ( K). n,∗ K n O Since ∆ /µΓ ( K) = ℓK[n] is finite, we conclude that it suffices to show the n,∗ K n O ∼ C analogue of Theorem 4 for individual elements M of ∆n,∗ K. This allows us to employ Theorem 1 b) and further restrict to fields generated by the entries of the matrices 1 A 0 ∆ in (3). u 0 D ∈ n,∗ K Now, recall from, for example, Section 3.1 of [3] that the Hilbert class field L of K is defined as the maximal abelian unramified extension of K and we may assume L C ⊆ by fixing an embedding. L has the property that every ideal K yields a principal A⊆ O ideal L L. More precisely, L K is a Galois extension with Galois group canoni- AO ⊆ O ⊇ cally isomorphic to ℓK. The n-torsion subgroup of ℓK via the Galois correspondence C C gives rise to an intermediate extension

L L[n] K ⊇ ⊇ with the property that an ideal K yields a principal ideal L L if and only A⊆ O AO [n] ⊆ O [n] [ ] ℓK[n]. This can be inferred from, for instance, the unique factorization property A ∈C

13 of ideals in K and the splitting behavior over L of prime ideals in K, which is outlined O O in [3]. n n Choose an ideal K satisfying [ ] ℓK[n] and let A × be a matrix A ⊆ O A ∈ C ∈ OK with elementary divisors ,..., as in the proof of Theorem 1. Then there is an A A n element u L such that L = u L . Set v = det A/u . Then we ′ ∈ O [n] AO [n] ′O [n] ′ ∈ OL∗ [n] have un = det A with u = u √n v, and therefore 1 A 0 ∆ for D, again, as in the ′ u 0 D ∈ n,∗ K proof of Theorem 1. In particular, we have found a preimage of [ ] ℓK[n] under the A ∈Cn homomorphism in Theorem 1 b), whose entries are contained in L[n]( √v). n To finish the proof, we have to show that L[n]( √v) Q is unramified outside of ndK. ⊇ As L K is unramified, so is L[n] K. If p K is a prime ideal that does not divide n, ⊇ ⊇ ⊆ O then the polynomial Xn v is separable modulo p, since its derivative nXn 1 does not − − vanish modulo p and neither does v.

We turn to the computation of representatives of ∆ /µΓ ( K), which we have im- n,∗ K n O plemented based on the computer algebra package Hecke [8]. By virtue of the proof of 1 both Theorem 1 and 4, it is naturally reduced to finding matrices u A of determinant 1 n n with A × and elementary divisors ,..., for [ ] ℓK[n]. The computation of ∈ OK A A A ∈C such A is achieved by the function hermitian extension in [23]. The two key aspects of hermitian extension are the use of the Hilbert class field in [23] and the function elementary divisor matrix. The latter produces an n n × matrix A over K with elementary divisors ,..., . To this end, we employ pseudo- O A A matrices from Definition 1.4.5 of [3]. Specifically, the pseudo-matrix A = (I, ( ,..., )) A A trivially has the desired elementary divisors. The Steinitz form of A is a pseudo- matrix A′, whose associated ideals are trivial, save the last one, whiche equals the Steinitz class of the ideals associated with A (cf. Theorem 1.2.12 of [3]). Since [e ] ℓK[n] by A ∈ C assumption,e this Steinitz class is principal and yields the desired matrix A. We conclude this paper with a briefe excerpt of discriminants of the fields generated by the entries of ∆n,∗ K, which we have obtained through our script. Observe that the powers of the discriminant of K originate in the Hilbert class field.

b Table 1: Discriminants dKn of the ﬁelds generated by the entries of ∆n,∗ K given n and K = Q(√ m) −

m −23 −31 −59 −83 −39 −55 −56 −68 −84 n 3 4 14 3 14 3 14 3 14 3 72 16 16 72 16 16 44 8 48 8 32 8 8 db Kn 3 23 3 31 3 59 3 83 2 3 13 2 5 11 2 7 2 17 2 3 7

Acknowledgement. The authors thank Joana Rodriguez for the direct proof of Theorem 2 and Claus Fieker for his help with Hecke [8].

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