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Multiplier systems for Hilbert’s and Siegel’s modular groups. (English) Zbl 0576.10021
For 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 \(n\geq 8)\).
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
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References:
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[2] Freitag, Discrete subgroups of Lie groups and applications to moduli pp 9– (1975)
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