We prove an equidistribution of signs for the Fourier coefficients of Hilbert modular forms of half-integral weight. Our study focuses on certain subfamilies of coefficients that are accessible via the Shimura correspondence. This is a generalization of the result of Inam and Wiese (Arch Math (Basel) 101:331-339, 2013) to the setting of totally real number fields.
Suppose E is an elliptic curve over Q of conductor N with complex multiplication (CM) by Q(i), and f E is the corresponding cuspidal Hecke eigenform in S new 2 (Γ 0 (N )). Then n-th Fourier coefficient of f E is non-zero in the short interval (X, X + cX 1 4 ) for all X ≫ 0 and for some c > 0. As a consequence, we produce infinitely many cuspidal CM eigenforms f level N > 1 and weight k > 2 for which i f (n) ≪ n 1 4 holds, for all n ≫ 0.
In this article, we study the simultaneous sign changes of the Fourier coefficients of two Hilbert cusp forms of different integral weights. We also study the simultaneous non-vanishing of Fourier coefficients, of two distinct non-zero primitive Hilbert cuspidal non-CM eigenforms of integral weights, at powers of a fixed prime ideal.
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