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