A Siegel cusp form of degree 12 and weight 12. 

J. reine angew. Math 494 (1998) 141-153. 

Richard E. Borcherds,
D.P.M.M.S., 16 Mill Lane, Cambridge, CB2 1SB, England. 

E. Freitag, 
Universit\"at Heidelberg, Im Neuenheimer Feld 288, D-69120 Heidelberg, 
Germany. 

R. Weissauer, 
Universit\"at Mannheim, D-68131 Mannheim, Germany. 

It has been conjectured by Witt [Wi] (1941) and proved later (1967) 
independently by Igusa [I] and M. Kneser [K] that the theta series with 
respect to the two unimodular even positive definite lattices of rank 16 are 
linearly dependent in degree \le 3 and linearly independent in degree 4. 
In this paper we consider the next case of the 24 Niemeier lattices of 
rank 24. The associated theta series are linearly dependent in degree 
\le 11 and linearly independent in degree 12. The resulting Siegel cusp 
form of degree 12 and weight 12 is a Hecke eigenform which seems to have 
interesting properties. We would like to thank G. H\"ohn for helpful comments 
and hints.