We present the GALEX UV photometry of the elliptical galaxies in Abell clusters at moderate redshifts (z < 0.2) for the study of the look-back time evolution of the UV upturn phenomenon. The brightest elliptical galaxies (M_r < -22) in 12 remote clusters are compared with the nearby giant elliptical galaxies of comparable optical luminosity in the Fornax and Virgo clusters. The sample galaxies presented here appear to be quiescent without signs of massive star formation or strong nuclear activity, and show smooth, extended profiles in their UV images indicating that the far-UV (FUV) light is mostly produced by hot stars in the underlying old stellar population. Compared to their counterparts in nearby clusters, the FUV flux of cluster giant elliptical galaxies at moderate redshifts fades rapidly with ~ 2 Gyrs of look-back time, and the observed pace in FUV - V color evolution agrees reasonably well with the prediction from the population synthesis models where the dominant FUV source is hot horizontal-branch stars and their progeny. A similar amount of color spread (~ 1 mag) in FUV - V exists among the brightest cluster elliptical galaxies at z ~ 0.1, as observed among the nearby giant elliptical galaxies of comparable optical luminosity.
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