Multiplier systems for Hilbert’s and Siegel’s modular groups.

*(English)*Zbl 0576.10021For the modular group SL(2,\({\mathbb{Z}})\) and its subgroups G, ”classical automorphy factors” of the form \(\nu (M)(cz+d)^ r\) of weight r (in \({\mathbb{C}})\) for complex z with \(Im(z)>0\) and \(M=\left( \begin{matrix} *\\ c\end{matrix} \begin{matrix} *\\ d\end{matrix} \right)\) in G and associated multiplier systems \(\{\) \(\nu\) (M)\(\}\) for the chosen branch of \(\log (cz+d)\) are known to exist for any complex weight r.

The situation is however different, when one considers instead, variables in generalized upper half planes \({\mathfrak H}_ n\) and groups \(\Gamma\) such as the Hilbert modular group \(\Gamma_ K\) (for a totally real number field K of degree m over \({\mathbb{Q}})\) or the Siegel modular group \(\Gamma_ n\) of degree n. In the case of \(\Gamma_ K\) for \(K={\mathbb{Q}}(\sqrt{5})\) or its theta subgroup, Maass showed that multiplier systems can exist only for integral or half-integral weight. Christian showed that, for subgroups (of finite index) in the Hilbert-Siegel modular group with m or \(n>1\), multiplier systems can arise only for rational weights; indeed for \(\Gamma_ n\), the weights are necessarily in \({\mathbb{Z}}\). The author showed that for \(K={\mathbb{Q}}(\sqrt{2})\), multiplier systems for \(\Gamma_ K\) have only integral weights; Endres proved that for the theta subgroup of \(\Gamma_ n\) with \(n>1\), the weights are half-integral. The principal strategy was to exploit the presence of (non-trivial) units for \(m>1\) or use clever matrix relations, in order to find a natural number g such that the multiple gr of the weight r is integral; this worked satisfactorily in special cases.

In this paper, the author finds a natural method uniformly applicable to all subgroups of finite index in \(\Gamma_ K\) or \(\Gamma_ n\) (with m or \(n>1)\); using carefully selected injections of the complex upper half- plane \({\mathfrak H}\) into \({\mathfrak H}_ n\) and associated imbeddings into \(\Gamma\) of groups \(\Lambda\) conjugate to congruence subgroups of SL(2,\({\mathbb{Z}})\) and further invoking an explicit congruence relation of Petersson for the weights of a multiplier system on \(\Lambda\) induced from the given multiplier system of weight r on \(\Gamma\), the author obtains a bound for the denominator of r, which is explicitly described for the case of the Hilbert modular group.

As a by-product of the rationality of the weights r, the values of the multipliers are shown to be necessarily roots of unity, when \(n>1\), upholding an assertion of Grosche (whose proof of the assertion is shown to have a gap). As remarked by the author, the non-existence of multiplier systems of weight r with 2r\(\not\in {\mathbb{Z}}\) can have nice applications (e.g. its use by Endres in the case of the theta subgroup of \(\Gamma_ n\) for \(n>1\), for concluding the irreducibility of the divisor of the classical theta function for \(n\geq 8)\).

The situation is however different, when one considers instead, variables in generalized upper half planes \({\mathfrak H}_ n\) and groups \(\Gamma\) such as the Hilbert modular group \(\Gamma_ K\) (for a totally real number field K of degree m over \({\mathbb{Q}})\) or the Siegel modular group \(\Gamma_ n\) of degree n. In the case of \(\Gamma_ K\) for \(K={\mathbb{Q}}(\sqrt{5})\) or its theta subgroup, Maass showed that multiplier systems can exist only for integral or half-integral weight. Christian showed that, for subgroups (of finite index) in the Hilbert-Siegel modular group with m or \(n>1\), multiplier systems can arise only for rational weights; indeed for \(\Gamma_ n\), the weights are necessarily in \({\mathbb{Z}}\). The author showed that for \(K={\mathbb{Q}}(\sqrt{2})\), multiplier systems for \(\Gamma_ K\) have only integral weights; Endres proved that for the theta subgroup of \(\Gamma_ n\) with \(n>1\), the weights are half-integral. The principal strategy was to exploit the presence of (non-trivial) units for \(m>1\) or use clever matrix relations, in order to find a natural number g such that the multiple gr of the weight r is integral; this worked satisfactorily in special cases.

In this paper, the author finds a natural method uniformly applicable to all subgroups of finite index in \(\Gamma_ K\) or \(\Gamma_ n\) (with m or \(n>1)\); using carefully selected injections of the complex upper half- plane \({\mathfrak H}\) into \({\mathfrak H}_ n\) and associated imbeddings into \(\Gamma\) of groups \(\Lambda\) conjugate to congruence subgroups of SL(2,\({\mathbb{Z}})\) and further invoking an explicit congruence relation of Petersson for the weights of a multiplier system on \(\Lambda\) induced from the given multiplier system of weight r on \(\Gamma\), the author obtains a bound for the denominator of r, which is explicitly described for the case of the Hilbert modular group.

As a by-product of the rationality of the weights r, the values of the multipliers are shown to be necessarily roots of unity, when \(n>1\), upholding an assertion of Grosche (whose proof of the assertion is shown to have a gap). As remarked by the author, the non-existence of multiplier systems of weight r with 2r\(\not\in {\mathbb{Z}}\) can have nice applications (e.g. its use by Endres in the case of the theta subgroup of \(\Gamma_ n\) for \(n>1\), for concluding the irreducibility of the divisor of the classical theta function for \(n\geq 8)\).

Reviewer: S.Raghavan

##### MSC:

11F27 | Theta series; Weil representation; theta correspondences |

11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |

32N05 | General theory of automorphic functions of several complex variables |

##### Keywords:

factors of automorphy; Hilbert modular group; Siegel modular group; multiplier systems; weights; subgroups of finite index
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DOI

##### References:

[1] | Grosche, Acta Arith. 33 pp 187– (1977) |

[2] | Freitag, Discrete subgroups of Lie groups and applications to moduli pp 9– (1975) |

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