Modular Moonshine II. 24 July 1994, corrected 19 Sept 1995 

Duke Math. J. 83 (1996) no. 2, 435-459. 

Richard E. Borcherds,
Mathematics department, University of California at Berkeley, CA 94720-3840 
U. S. A. 

Alex J. E. Ryba, Dept of Mathematics, Marquette University, Milwaukee, 
WI 53233, U. S. A. 

The monster simple group acts on the monster vertex algebra, and the 
moonshine conjectures state that the traces of elements of the monster 
on the vertex algebra are Hauptmoduls. Ryba [R94] conjectured the 
existence of similar vertex algebras over fields of characteristic p 
acted on by the centralizers of certain elements of prime order p in 
the monster, and conjectured that the Brauer traces of p-regular elements 
of the centralizers were certain Hauptmoduls. We will prove these 
conjectures when the centralizer involves a sporadic group (p \le 11, 
corresponding to the sporadic groups B, Fi'_{24}, Th, HN, He, and M_{12}).