Abstract
We derive an explicit isomorphism between the Hilbert modular group and certain congruence subgroups on the one hand and particular subgroups of the special orthogonal group SO(2, 2) on the other hand. The proof is based on an application of linear algebra adapted to number theoretical needs.
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1 Introduction
A generalization of the classical elliptic moduar group \(SL_2(\mathbb {Z})\) was introduced by Blumenthal [2, 3] more than 100 years ago. \(SL_2({\scriptstyle {\mathcal{O}}}_\mathbb {K})\) is the ordinary Hilbert modular group over a totally real number field \(\mathbb {K}\). Meanwhile the theory of Hilbert modular forms has been developped into various directions (cf. [6, 7, 18]). On the other hand Borcherds [4] established a product expansion for modular forms on SO(2, n). If \(n=2\) and \(\mathbb {K}\) is a real-quadratic field these groups are basically isomorphic. More precisely \(PSL_2({\scriptstyle {\mathcal{O}}}_\mathbb {K})\) is isomorphic to discriminant kernel of the orthogonal group SO(2, 2) (cf. Theorem 2). Borcherds theory has led to important examples of Hilbert modular forms (cf. [5]), which in some cases allow to describe the graded ring of Hilbert modular forms (cf. [17, 19]). As Borcherds products are usually related with discriminant kernels, it is interesting to describe the associated congruence subgroups of \(PSL_2(\mathbb {R})\) precisely.
In this paper we derive an explicit isomophism based on linear algebra. Our approach can be extended to congruence subgroups. The results below can serve as an explicit dictionary, when passing between Borcherds theory and classical Hilbert modular forms.
Given a non-degenerate symmetric even matrix \(T\in \mathbb {Z}^{m\times m}\)let
denote the attached special orthogonal group. Let \(SO_0(T;\mathbb {R})\) stand for the connected component of the identity matrix I and \(SO_0(T;\mathbb {Z})\) for the subgroup of integral matrices. The discriminant kernel
is clearly a normal subgroup of \(SO_0(T;\mathbb {Z})\). Given \(N\in \mathbb {N}\) we set
for the orthogonal sum with the rescaled hyperbolic plane.
2 The normalizer of the Hilbert modular group
Throughout this paper let \(\mathbb {K}=\mathbb {Q}(\sqrt{m})\), \(m\in \mathbb {N}\), \(m>1\) squarefree, be a real-quadratic number field with ring of integers and discriminant
The non-trivial automorphism of \(\mathbb {K}\) is given by
If \(\ell \in \mathbb {N}\), \(\sqrt{\ell \,}\notin \mathbb {K}\) we extend (4) to \(\mathbb {K}(\sqrt{\ell \,})\) via \(\sqrt{\ell \,}' = \sqrt{\ell \,}\). Moreover \(a\gg 0\) means that \(a\in \mathbb {K}\) is totally positive, i.e. \(a>0\) and \(a'>0\).
The (ordinary) Hilbert modular group (cf. [2, 3, 6]) is given by
At first we are going to describe the normalizer
of \(\Gamma _\mathbb {K}\) in \(SL_2(\mathbb {R})\). Therefore let \(\mathcal {I}(L)\) denote the fractional ideal generated by the entries of a matrix \(0\ne L \in \mathbb {K}^{2\times 2}\).
Theorem 1
If \(\mathbb {K}\) is a real-quadratic number field, the normalizer \(\mathcal {N}_\mathbb {K}\) is equal to
Proof
Clearly the matrices in (5) belong to \(\mathcal {N}_\mathbb {K}\). As \(\Gamma _\mathbb {K}\) contains a \(\mathbb {Z}\)-basis of \({\scriptstyle {\mathcal{O}}}^{2\times 2}_\mathbb {K}\), we conclude that any \(M\in \mathcal {N}_\mathbb {K}\) satisfies
Inserting the classical basis elements, we see that the product of any two entries of M belongs to \({\scriptstyle {\mathcal{O}}}_K\). If \(\alpha \) is any non-zero entry of M, we get
Thus M belongs to (5). \(\square \)
Now we define the group
Note that \(M_1, M_2 \in \Sigma _\mathbb {K}\) satisfy
due to the above extension of (4).
Let \(\mathcal {H}\) denote the upper half-plane in \(\mathbb {C}\). Then a matrix \(M=\left( {\begin{matrix} \alpha &{} \beta \\ \gamma &{} \delta \end{matrix}}\right) \in \Sigma _\mathbb {K}\) acts on \(\mathcal {H}^2\) via
It is well-known that a subgroup \(\Gamma \subseteq \Sigma _\mathbb {K}\) acts dicontinuously if and only if the embedded group
is discrete (cf. [6], I.2.1). The maximal discontinuous extension \(\Gamma ^*_\mathbb {K}\) of \(\Gamma _\mathbb {K}\) is called the Hurwitz–Maaß extension. It is described in [1, 9, 16] as
where we define the generalized Atkin-Lehner matrices by
If \(\mathcal {A}_\ell = \mathbb {Z}\ell + \mathbb {Z}\omega _\mathbb {K}\) denotes the integral ideal in \({\scriptstyle {\mathcal{O}}}_\mathbb {K}\) of reduced norm \(\ell \) for a sqaurefree divisor \(\ell \) of \(d_\mathbb {K}\), we have
Note that for squarefree divisors \(k,\ell \) of \(d_\mathbb {K}\), we have
Note that due to (8) and Theorem 1, in some cases there exists a matrix \(M_0\in \mathcal {N}_\mathbb {K}\), \(M_0\notin \Gamma ^*_\mathbb {K}\). If the fundamental unit \(\varepsilon _0\) satisfies \(\varepsilon _0\varepsilon '_0 = -1\), we may choose
If \(m= \alpha ^2+\beta ^2\) for some \(\alpha ,\beta \in \mathbb {N}\), \(\alpha \) odd, we choose \(\nu ,\mu \in \mathbb {Z}\) satisfying \(\nu \alpha -2\mu \beta = 1\), \(u: = \beta +\sqrt{m} \in {\scriptstyle {\mathcal{O}}}_\mathbb {K}\), \(uu' = -\alpha ^2\),
Thus we may choose
in this case. If we apply the theory of ambiguous ideals, then [15], 7.6 and 7.8, imply
Corollary 1
Let \(\mathbb {K}\) be a real-quadratic number field with fundamental unit \(\varepsilon _0\). If \(\varepsilon _0 \varepsilon '_0 = 1\) and \(p\mid d_\mathbb {K}\) for some prime \(p\equiv 3\bmod {4}\), we have
In any other case
holds with \(M_0\) from (11) and (12).
We can quote Maaß [16] or apply the results from (8) and (10) in order to obtain
Remark 1
a) The field \(\widehat{\mathbb {K}}\) generated by the entries of the matrices in \(\Gamma ^*_\mathbb {K}\) is given by
according to (8) and (9). Hence an analog of Theorem 4 in [13] also holds in this case:
b) If \(\varepsilon \gg 0\) is a unit in \({\scriptstyle {\mathcal{O}}}_\mathbb {K}\), then (8) yields
Thus the squarefree kernel q of \(2+\varepsilon + \varepsilon '\) always divides \(d_\mathbb {K}\).
3 The Hilbert modular group as an orthogonal group
If \(T\in \mathbb {Z}^{2\times 2}\) is an even symmetric matrix with \(\det T< 0\) the associated half-space \(\mathcal {H}_T\) is defined to be
Given \(\widetilde{M}\in SO_0(T_N;\mathbb {R})\) (cf. (2)) we will always assume the form
It is well-known (cf. [8, 12]) that \(\widetilde{M}\) acts on \(\mathcal {H}_T\) via
Note that \(\mathcal {H}^2\), where \(\mathcal {H}\) denotes the upper half-plane in \(\mathbb {C}\), is the orthogonal half-space \(\mathcal {H}_P\), \(P=\left( {\begin{matrix} 0 &{} 1 \\ 1 &{} 0 \end{matrix}}\right) \). Considering [12], Sect. 5, we obtain a surjective homomorphism of the groups
with kernel \(\{\pm (I,I)\}\) satisfying
Thus we have for \(M\in \Sigma _\mathbb {K}\)
Note that \(\Omega (M,M') \ne -I\) for all these M. Now choose a basis \(B=(u,v)\) of \(\mathbb {K}\) over \(\mathbb {Q}\) and consider the base change
We set
If we define
a straightforward calculation yields that the description of \(\widetilde{M}\) in (13) is given by
Recalling the definition of the discriminant kernel from (1), we obtain
Theorem 2
Let \(\mathbb {K}\) be a real-quadratic field and let \(B=(u,v)\) be a \(\mathbb {Z}\)-basis of \({\scriptstyle {\mathcal{O}}}_\mathbb {K}\) with Gram matrix S from (17). Then the mappings
where \(\widetilde{M}\) is defined by (18), are isomorphisms of the groups. Moreover the mapping
is an isomorphism of the groups.
Proof
We get \(\widetilde{\phi }_B\bigl (\Sigma _\mathbb {K}/\{\pm I\}\bigr ) \subseteq SO_0(S_1;\mathbb {Q})/\{\pm I\}\) and \(\widetilde{\phi }_B\bigl (\Gamma ^*_\mathbb {K}/\{\pm I\}\bigr ) \subseteq SO_0(S_1;\mathbb {Z})/\{\pm I\}\) directly from (18). As \(\widetilde{\phi }_B\) maps the group actions in (14) and (15) onto each other, \(\widetilde{\phi }_B\) is an injective homomorphism of the groups. In the notation of [12] we easily see that \(SO_0(S_1;\mathbb {Q})\) is generated by the matrices
They appear as images of the matrices
in \(\Sigma _\mathbb {K}\). Hence \(\widetilde{\phi }_B\) is an isomorphism for the rationals. As \(\widetilde{\phi }_B^{-1}\bigl (SO_0(S_1;\mathbb {Z})/\{\pm I\}\bigr )\) is a discontinuous subgroup of \(PSL_2(\mathbb {R})\) containing \(\Gamma ^*_\mathbb {K}/\{\pm I\}\), we obtain equality from the result of Maaß [16].
In view of \(-I\not \in \phi _B\bigl (\Gamma _\mathbb {K}/\{\pm I\}\bigr )\) we conclude that \(\phi _B\) is an injective homomophism of the groups with \(\phi _B\bigl (\Gamma _\mathbb {K}/\{\pm I\}\bigr )\subseteq SO_0(S_1;\mathbb {Z})\). We get
if and only if \(F_M = f_M -id\) satisfies
If \(X\in \mathbb {Z}^{2\times 2}\) satisfies
the latter condition becomes equivalent to
as
Thus (19) holds if and only if
This is easily demonstrated for \(M\in \Gamma _\mathbb {K}\) using \(\alpha \delta -\beta \gamma = 1\). If \(\ell \) is a squarefree divisor of \(d_\mathbb {K}\), \(\ell \ne 1,m\) we verify
Thus \(\phi _B\) becomes an isomorphism, too. \(\square \)
Now we apply the results to congruence subgroups.
Corollary 2
Let \(N\in \mathbb {N}\) and \(B=(u,v)\) be a \(\mathbb {Z}\)-basis of \({\scriptstyle {\mathcal{O}}}_\mathbb {K}\) with Gram matrix S from (17). (a) \(\phi _B \) maps
onto
(b) \(\phi _B\) maps the principal congruence subgroup
onto
Proof
(b) Given M in the principal congruence subgroup, then \(\phi _B(\pm M) \in \mathcal {D}(NS_1;\mathbb {Z})\) is a consequence of (18) and Theorem 2. On the other hand \(\pm M\in \phi _B^{-1}\left( \mathcal {D}(NS_1;\mathbb {Z})\right) \) implies \(\beta ,\gamma \in N{\scriptstyle {\mathcal{O}}}_\mathbb {K}\) due to (18). Considering the representing matrix of \(w\mapsto \alpha \delta ' w\) and using \(\alpha \delta \equiv 1 \bmod {N{\scriptstyle {\mathcal{O}}}_\mathbb {K}}\) we obtain
\(\square \)
We add a Remark
Remark 2
\({\scriptstyle {\mathcal{O}}}_\mathbb {K}\) with the symmetric bilinear form \((\alpha , \beta )\mapsto \alpha \beta ' + \alpha '\beta \) is a maximal even lattice. Thus Theorem 2 corresponds to the results on SO(2, n), \(n\ge 3\) in [14].
4 The general Hilbert modular group
Given an integral ideal \(0\ne \mathcal {I}\subseteq {\scriptstyle {\mathcal{O}}}_\mathbb {K}\) the (general) Hilbert modular group with respect to \(\mathcal {I}\) is defined by
Given a basis B of \(\mathbb {K}\) then Theorem 2 and (18) immediately imply
whenever \(n\in \mathbb {N}\) and \(H_n = \phi _B\left( \pm \frac{1}{\sqrt{n}}\left( {\begin{matrix} n &{} 0 \\ 0 &{} 1 \end{matrix}}\right) \right) ={\text {diag}}\,(n,1,1,1/n)\). Hence we may assume that \(\mathcal {I}\) is a primitive ideal, i.e.
In this case \(\mathcal {I}\) contains a \(\mathbb {Z}\)-basis
where N is the reduced norm of \(\mathcal {I}\) (cf. [15], Proposition 4.2). It follows from [9, 10] or [16] that the so-called Maaß-Hurwitz extension \(\Gamma ^*_\mathbb {K}(\mathcal {I})\) is a maximal discontinuous subgroup of \(SL_2(\mathbb {R})\) containing \(\Gamma _\mathbb {K}(\mathcal {I})\) with index
It is given by
One may choose
Just as in Sect. 2 we have
as well as
Theorem 3
Let \(\mathbb {K}\) be a real-quadratic field and let \(\mathcal {I}\subseteq {\scriptstyle {\mathcal{O}}}_\mathbb {K}\) be a primitive ideal with reduced norm N. Assume that \(B = (u,v)\) is a \(\mathbb {Z}\)-basis of \(\mathcal {I}\) with Gram-matrix S from (17) and let \(T = \frac{1}{N} S\). Then the mappings
where \(H_N={\text {diag}}\,(N,1,1,1)\) and \(\widetilde{M}\) is given by (18), are isomorphisms of the groups.
Proof
Proceed exactly as in the proof of Theorem 2. Observe that \(\mathcal {I}^{-1} = \frac{1}{N} \mathcal {I}'\) and verify that
Use the basis (20) of \(\mathcal {I}\) in order to verify that
whenever \(\ell \) is a squarefree divisor of \(d_\mathbb {K}\).
We obtain an immediate application to congruence subgroups of \(\Gamma _\mathbb {K}\).
Corollary 3
Let N be a squarefree divisor of \(d_\mathbb {K}\). Then the principal congruence subgroup
is isomorphic to
via \(\phi _B\), where \(B = (N,\omega _\mathbb {K})\).
Proof
Apply Theorem 3 to \(\mathcal {I}= \mathcal {A}_N = \mathbb {Z}N + \mathbb {Z}\omega _\mathbb {K}\) and follow the proof of Corollary 2 a). \(\square \)
We add a final Remark.
Remark 3
The congruence conditions with respect to N as opposed to more general \(\mathcal {I}\) in Corollaries 2 and 3 are equivalent to the restriction to ideals \(\mathcal {I}\) satisfying \(\mathcal {I}= \mathcal {I}'\). It is not only motivated by technical reasons, but also by the fact that symmetric Hilbert modular forms, i.e. \(f(\tau _2,\tau _1) =f(\tau _1,\tau _2)\), can be defined for these subgroups.
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Hauffe-Waschbüsch, A., Krieg, A. The Hilbert modular group and orthogonal groups. Res. number theory 8, 47 (2022). https://doi.org/10.1007/s40993-022-00346-5
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DOI: https://doi.org/10.1007/s40993-022-00346-5