Reflection groups of Lorentzian lattices. 21 September 1999. 
Duke Math. J. 104 (2000) no. 2, 319-366. 
Richard E. Borcherds, 
Math Dept, Evans Hall #3840, University of California at Berkeley, 
CA 94720-3840, U.S.A. 
home page: http://math.berkeley.edu/~reb 

Contents.  
1. Introduction. 
Notation. 
2. Modular forms. 
3. Discriminant forms and the Weil representation. 
4. The singular theta correspondence. 
5. Theta functions. 
6. Eta quotients. 
7. Dimensions of spaces of modular forms. 
8. The geometry of G0(N). 
9. An application of Serre duality. 
10. Eisenstein series. 
11. Reflective forms. 
12. Examples. 
13. Open problems. 

1.  Introduction. 

The aim of this paper is to provide evidence for the following new principle: 
interesting reflection groups of Lorentzian lattices are controlled by certain 
modular forms with poles at cusps. We use this principle to explain many of 
the known examples of such reflection groups, and to find several new examples 
of reflection groups of Lorentzian lattices, including one whose fundamental 
domain has 960 faces.