Mayall II = G1 in M31: Giant Globular Cluster or Core of a Dwarf Elliptical Galaxy ?

(Abridged version) Mayall II = G1 is one of the brightest globular clusters belonging to M31, the Andromeda galaxy. Our observations with HST/WFPC2 provide data for the (I vs. V-I) and (V vs. V-I) color-magnitude diagrams. From model fitting, we determine a rather high mean metallicity of [Fe/H] = --0.95 +- 0.09, somewhat similar to 47 Tucanae. We find a larger spread in V-I than can be explained by the measurement errors, and we attribute this to an intrinsic metallicity dispersion amongst the stars of G1. So far, only omega Centauri, the giant Galactic globular cluster, has been known to exhibit such an intrinsic metallicity dispersion. Three estimates of the total mass of this globular cluster can be obtained: King mass = 15 x 10^6 with M/Lv ~ 7.5, Virial mass = 7.3 x 10^6 with M/Lv ~ 3.6, and King-Michie mass range from 14 to 17 x 10^6. Although uncertain, all of these mass estimates make G1 more than twice as massive as omega Centauri. Such large masses relate to the metallicity spread whose origin is still unknown (either self-enrichment, an inhomogeneous proto-cluster cloud, or remaining core of a dwarf galaxy). When considering the positions of G1 in the different diagrams defined by Kormendy (1985), G1 always appears on the sequence defined by globular clusters, and definitely away from the other sequences defined by elliptical galaxies, bulges, and dwarf spheroidal galaxies. The same is true for omega Centauri and for the nucleus of the dwarf elliptical NGC 205. This does not prove that all (massive) globular clusters are the remnant cores of nucleated dwarf galaxies.