On the Fourier coefficients of Hilbert modular forms of half-integral weight over arbitrary algebraic number fields

We denote by Z, Q, R and C the ring of rational integers, the rational number field, the real number field and the complex number field, respectively. We write F for an algebraic number field, d for the different of F relative to Q, o for the integral ring of F . F has r1 real archimedian primes and r2 imaginary archimedian primes. σi : F → R (1 ≤ i ≤ r1) are the mutually distinct embeddings of F to R, and σr1+j : F → C (1 ≤ j ≤ r2) are the mutually distinct imaginary conjugate embeddings of F to C such that σr1+j 6= σr1+l, σr1+j 6= σr1+l (1 ≤ j, l ≤ r2, j 6= l), σr1+j 6= σr1+j (1 ≤ j ≤ r2) and σi 6= σr1+j (1 ≤ i ≤ r1, 1 ≤ j ≤ r2). For α ∈ F , we put α = σi(α) and α (r1+j) = σr1+j(α) (1 ≤ i ≤ r1, 1 ≤ j ≤ r2). Let H = R + Ri + Rj + Rk be the Hamilton quaternion algebra, H = {z = z + wj ∈ H|z ∈ C, w > 0}, H = {z = x + iy|x ∈ R, y > 0} and D = H1 × H2 .