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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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