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On multiplier systems and theta functions of half-integral weight for the Hilbert modular group \(\operatorname{SL}_2(\mathfrak{o})\). (English) Zbl 1522.11036

The author computes the characters of the metaplectic group \(\widetilde{\mathrm{SL}_2(\mathfrak{o})}\) over the ring of integers \(\mathfrak{o}\) of a \(p\)-adic number field. As a corollary, it is shown that the level one Hilbert modular group \(\mathrm{SL}_2(\mathfrak{o}_F)\) attached to a totally real number field \(F\) admits multiplier systems of half-integral weight if and only if the prime \(2\) splits completely in \(F\). In the latter case, the author classifies theta functions that are Hilbert modular forms of weights \(1/2\) and \(3/2\) in terms of fractional ideals.

MSC:

11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F27 Theta series; Weil representation; theta correspondences
11F37 Forms of half-integer weight; nonholomorphic modular forms
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References:

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