In this paper, we first prove an isomorphism between certain spaces of Jacobi forms.
Using this isomorphism, we study the mod p theory of Hermitian Jacobi forms over {\mathbb{Q}(i)}.
We then apply the mod p theory of Hermitian Jacobi forms to characterize {U(p)} congruences and to study Ramanujan-type congruences for Hermitian Jacobi forms and Hermitian modular forms of degree 2 over {\mathbb{Q}(i)}.
We prove a quantitative result for the number of sign changes of the Fourier coefficients of a Hermitian cusp form of degree 2. In addition, we prove a quantitative result for the number of sign changes of the primitive Fourier coefficients. We give an explicit upper bound for the first sign change of the Fourier coefficients of a Hermitian cusp form of degree 2 over certain imaginary quadratic extensions.
We obtain some interesting results about the parity of the Fourier coefficients of hauptmoduln j N (z) and j + N (z), for some positive integers N . We use elementary methods and the techniques of O. Kolberg's proof for the parity of the partition function.
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