We present the results of deep imaging of the field of GRB 050904 with Suprime-Cam on the Subaru 8.2 m telescope. We have obtained a narrow-band (130 Å) image centered at 9200 Å (NB921) and an i′-band image with total integration times of 56700 and 24060 s, respectively. The host galaxy was not detected within 1″ of the afterglow position. An object was found at 1.5″ NE from the position of the afterglow, but clear detection of this object in the i′-band image rules out its association with the burst. We obtained a limit of > 26.4 AB magnitude (2″ diameter, 3 σ) in the NB921 image for the host galaxy, corresponding to a flux of 6.0 × 10 28 erg/s/Hz at rest 1500 Å assuming a flat spectrum of the host galaxy. The star formation rate should be less than 7.5 (M⊙/y) based on the conversion rate by Madau, Pozzetti and Dickinson (Astrophys. J., 498 (1998) 106). This upper limit for the host of GRB 050904 is consistent with the star formation rate of other gamma-ray burst host galaxies around redshift of 2 or less.
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