Hermitian modular forms congruent to 1 modulo p

TL;DR

This paper constructs Hermitian modular forms congruent to 1 modulo p over imaginary quadratic fields, using theta series of lattices with automorphisms, extending to non-free lattices.

Contribution

It introduces a method to produce Hermitian modular forms congruent to 1 modulo p from lattices with specific automorphisms, including non-free lattices.

Findings

Existence of Hermitian modular forms congruent to 1 mod p for arbitrary genus

Construction of such forms using theta series of lattices with automorphisms

Extension of modularity to non-free lattices

Abstract

For any natural number $\ell $ and any prime $p\equiv 1 \pmod{4}$ 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.