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Let $F$ be a totally real number field and $\mathfrak{o}$ the ring ofintegers of $F$. We study theta functions which are Hilbert modular forms ofhalf-integral weight for the Hilbert modular group$\mathrm{SL}_2(\mathfrak{o})$. We obtain an equivalent condition that thereexists a multiplier system of half-integral weight for$\mathrm{SL}_2(\mathfrak{o})$. We determine the condition of $F$ that thereexists a theta function which is a Hilbert modular form of half-integral weightfor $\mathrm{SL}_2(\mathfrak{o})$. The theta function is defined by a sum on afractional ideal $\mathfrak{a}$ of $F$.

On multiplier systems and theta functions of half-integral weight for the Hilbert modular group $\mathrm{SL}_2(\mathfrak{o})$