We present the results of deep imaging of the field of GRB 050904 with Suprime-Cam on the Subaru 8.2m telescope .
We have obtained a narrow-band ( 130 Å ) image centered at 9200 Å ( NB921 ) and an i ^ { \prime } -band image with total integration times of 56700 and 24060 s , respectively .
The host galaxy was not detected within 1 ^ { \prime \prime } of the afterglow position .
An object was found at 1.5 ^ { \prime \prime } NE from the position of the afterglow , but clear detection of this object in the i ^ { \prime } -band image rules out its association with the burst .
We obtained a limit of > 26.4 AB magnitude ( 2 ^ { \prime \prime } diameter , 3 \sigma ) in the NB921 image for the host galaxy , corresponding to a flux of 6.0 \times 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 _ { \odot } /yr ) based on the conversion rate by [ 2 ] .
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 .