Holomorphic differential forms on some orthogonal modular varieties

Abstract

We construct holomorphic differential forms of many degrees, including the minimum possible one, on the modular varieties associated to the even lattices of signature $(2,n)$ with $n\equiv 1,3$ mod $8$ and discriminant $-2$ in the range $n\ge 25$. This is the first example of holomorphic differential forms of non-top degree on orthogonal modular varieties. The proof uses the Arthur multiplicity formula in the theory of automorphic representations.