Hermitian modular forms congruent to 1 modulo p

Abstract

For any natural number $\ell$ and any prime $p\equiv 1(\mathrm{mod}4)$ not dividing $\ell$ there is a Hermitian modular form of arbitrary genus $n$ over $L:=\Q [\sqrt{-\ell}]$ that is congruent to 1 modulo $p$ which is a Hermitian theta series of an $O_L$-lattice of rank $p-1$ admitting a fixed point free automorphism of order $p$. It is shown that also for non-free lattices such theta series are modular forms.