The brightest asymptotic giant branch stars in the inner bulge of M31

JHK images with angular resolutions approaching the diffraction limit of th e 3.6 m Canada-France-Hawaii Telescope are used to investigate the bright a symptotic giant branch (AGB) content of the M31 bulge. The AGB tip in a fie ld from the galaxy center occurs at K = 15.6, which is significantly fainte r than measured in previous ground-based studies that sampled similar proje cted distances from the center of M31 but were affected by crowding. Within of the center of M31, the number density of bright 2@.6 AGB stars scales w ith r-band surface brightness and the K brightness of the AGB tip does not vary measurably with radius. It is concluded that the infrared bright AGB s tars (1) belong to the bulge and not the disk and (2) are well mixed throug hout the inner bulge, suggesting that they formed at a time when the overal l structural properties of the M31 bulge were imprinted. The bolometric lum inosity functions (LFs) of the M31 bulge and Baade's window are in excellen t agreement, while the brightest AGB stars in the M31 bulge, the Galactic b ulge, and M32 have similar M-K. Barring a fortuitous tuning of age and meta llicity to produce AGB tips with similar brightnesses, it is suggested that the brightest stars in M32 and the bulges of M31 and the Milky Way belong to an old, metal-rich population; these stars are bright not because they h ave a young or intermediate age, but because they have a high metallicity.