Let F be a totally real number field and 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 SL2(o). We obtain an equivalent condition that there exists a multiplier system of half-integral weight for SL2(o). We determine the condition of F that there exists a theta function which is a Hilbert modular form of half-integral weight for SL2(o). The theta function is defined by a sum on a fractional ideal a of F .
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