Hermitian modular forms congruent to 1 modulo p

Abstract

For any natural number $$\ell$$ and any prime $$p \equiv 1$$ (mod 4) not dividing $$\ell$$ there is a Hermitian modular form of arbitrary genus n over $$L := {\mathbb{Q}}[\sqrt{-\ell}]$$ 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.