The maximal discrete extension of the Hermitian modular group by Aloys Krieg, Martin Raum and Annalena Wernz1 Abstract. Let Γ ( K) denote the Hermitian modular group of degree n over an n O imaginary-quadratic number field K. In this paper we determine its maximal discrete extension in SU(n,n; C), which coincides with the normalizer of Γ ( K). The descrip- n O tion involves the n-torsion subgroup of the ideal class group of K. This group is defined over a particular number field Kn and we can describe the ramified primes in it. In the case n = 2 we give an explicit description, which involves generalized Atkin-Lehner involutions. Moreover we find ab natural characterization of this group in SO(2, 4). Keywords. Hermitian modular group, normalizer, maximal discrete extension, Atkin- Lehner involution, orthogonal group Mathematics Subject Classification. 11F06, 11F55 1Aloys Krieg, Annalena Wernz, Lehrstuhl A f¨ur Mathematik, RWTH Aachen University, D-52056 Aachen, Germany arXiv:1910.12466v2 [math.NT] 27 Jan 2021 [email protected], [email protected] Martin Raum, Chalmers tekniska h¨ogskola och G¨oteborgs Universitet, Institutionen f¨or Matematiska vetenskaper, SE-412 96, G¨oteborg, Sweden [email protected] Martin Raum was partially supported by Vetenskapsr˚adet Grant 2015-04139. Annalena Wernz was partially supported by Graduiertenkolleg Experimentelle und konstruktive Al- gebra at RWTH Aachen University. 1 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 I H 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.