On the signs of Fourier coefficients of Hilbert cusp forms

Abstract

We prove that given any $\epsilon > 0$ and a primitive adelic Hilbert cusp form f of weight $k=(k_1,k_2,\ldots ,k_n) \in (2 {\mathbb {Z}})^n$ and full level, there exists an integral ideal ${\mathfrak {m}}$ with $N({\mathfrak {m}}) \ll _{\epsilon } Q_{f}^{9/20+ \epsilon }$ such that the ${\mathfrak {m}}$-th Fourier coefficient of $C_{f} ({\mathfrak {m}})$ of f is negative. Here n is the degree of the associated number field, $N({\mathfrak {m}})$ is the norm of integral ideal ${\mathfrak {m}}$ and $Q_{f}$ is the analytic conductor of f. In the case of arbitrary weights, we show that there is an integral ideal ${\mathfrak {m}}$ with $N({\mathfrak {m}}) \ll _{\epsilon } Q_{f}^{1/2 + \epsilon }$ such that $C_{f}({\mathfrak {m}}) <0$. We also prove that when $k=(k_1,k_2,\ldots ,k_n) \in (2 {\mathbb {Z}})^n$, asymptotically half of the Fourier coefficients are positive while half are negative.