论文标题
关于希尔伯特尖缘形式的傅立叶系数的迹象
On the signs of Fourier coefficients of Hilbert cusp forms
论文作者
论文摘要
我们证明，给定任何$ε> 0 $和原始的Adelic Hilbert cusp form $ f $ f $ f $ a r $ k =（k_1，k_2，...，...，k_n）\ in（2 \ mathbb {z} n $和完整级别，有一个整体级别的$ \ nathfrak $ n（Mathfrak $ n（l Math）， q_ {f}^{9/20+ε} $，使得$ \ mathfrak {m} $ - $ c_ {f}（\ mathfrak {m} $ f $ of $ f $ of $ c_ {f}（\ mathfrak {m Mathfrak {m Mathfrak {m Mathfrak）是负数。这里$ n $是关联的数字字段的程度，$ n（\ mathfrak {m}）$是积分理想$ \ mathfrak {m} $和$ q_ {f} $的整体标准的规范。在任意权重的情况下，我们表明有一个整体理想的$ \ mathfrak {m} $，带有$ n（\ mathfrak {m}）\ll_εq_ q_ Q_ {f}^{1/2 +ε} $，因此$ c_ {f}（f}（f}）我们还证明，当$ k =（k_1，k_2，...，k_n）\ in（2 \ mathbb {z}）^n $ in（渐近上一半的傅立叶系数是正的，一半是负的，而一半为负。
We prove that given any $ε> 0$ and a primitive adelic Hilbert cusp form $f$ of weight $k=(k_1,k_2,...,k_n) \in (2 \mathbb{Z})^n$ and full level, there exists an integral ideal $\mathfrak{m}$ with $N(\mathfrak{m}) \ll_ε Q_{f}^{9/20+ ε} $ 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 $N(\mathfrak{m}) \ll_ε Q_{f}^{1/2 + ε}$ such that $C_{f}(\mathfrak{m}) <0$. We also prove that when $k=(k_1,k_2,...,k_n) \in (2 \mathbb{Z})^n$, asymptotically half of the Fourier coefficients are positive while half are negative.
