Modular Moonshine III. 11 October 1994, corrected 18 Nov 1997 

Duke Math. J. 93 (1998), no. 1, 129-154. 

Richard E. Borcherds,

D.P.M.M.S., 16 Mill Lane, Cambridge, CB2 2SB, England. 
Mathematics department, University of California at Berkeley, 
CA 94720-3840, U. S. A. 

www.dpmms.cam.ac.uk/~reb 

In this paper we complete the proof of Ryba's modular moonshine conjectures 
[R] that was started in [B-R]. We do this by applying Hodge theory to the 
cohomology of the monster Lie algebra over the ring of p-adic integers in 
order to calculate the Tate cohomology groups of elements of the monster 
acting on the monster vertex algebra.