Congruence properties of Hermitian modular forms

Abstract

Abstract. We study the existence of a modular form satisfying a certain congruence relation. The existence of such modular forms plays an important role in the determination of the structure of a ring of modular forms modulo p. We give a criterion for the existence of such a modular form in the case of Hermitian modular forms.

Hermitian modular forms congruent to 1 modulo p

“…Together this implies the existence of a Hermitian modular form of weight p − 1 that is congruent to 1 modulo p for more general imaginary quadratic number fields than those treated in [11]:…”

“…The purpose of the present note is to generalize the construction of Siegel modular forms that are congruent to 1 modulo a suitable prime p given in [3] to the case of Hermitian modular forms over L := Q[ √ −ℓ]. For ℓ = 1 and ℓ = 3 this was done in [11], in fact we use the same strategy by constructing an even unimodular lattice Λ as an ideal lattice in K := L[ζ p ] for any prime p ≡ 1 (mod 4) not dividing ℓ. The existence of Λ essentially follows from class field theory and is predicted by [2, Théorème 2.3, Proposition 3.1 (1)] (see also [1,Corollary 2]).…”