论文标题
关于Hilbert cusp形式的傅立叶系数的大小
On the size of the Fourier coefficients of Hilbert cusp forms
论文作者
论文摘要
令$ \ bf f $是一种原始的希尔伯特风格形式的重量$ k $和级别$ \ mathfrak {n} $，带有傅立叶系数$ c _ {\ bf f}（\ mathfrak {m mathfrak {m}）$。我们证明了几乎所有傅立叶系数的非平凡的上限$ c _ {\ bf f}（\ mathfrak {m}）$ \ bf f $。这概括了Luca，Radziwił和Shparlinski获得的界限。我们还证明了无限的许多积分理想的存在$ \ mathfrak {m} $，傅立叶系数$ c _ {\ bf f}（\ mathfrak {m Mathfrak {m}）$具有改进的上限，我们进一步获得了这些积分理想的精制，从而使这些整体理想在Prime Powers方面得到了改进。特别是，这使我们能够推断出椭圆形尖的傅立叶系数以外的“典型尺寸”。此外，我们证明了在利特伍德猜想的假设下的界限进一步改善。最后，如果相应的Hecke特征角度差异很差，我们研究了Prime功率处的傅立叶系数的下限。
Let $\bf f$ be a primitive Hilbert cusp form of weight $k$ and level $\mathfrak{n}$ with Fourier coefficients $c_{\bf f}(\mathfrak{m})$. We prove a non-trivial upper bound for almost all Fourier coefficients $c_{\bf f}(\mathfrak{m})$ of $\bf f$. This generalizes the bounds obtained by Luca, Radziwiłł and Shparlinski. We also prove the existence of infinitely many integral ideals $\mathfrak{m}$ for which the Fourier coefficients $c_{\bf f}(\mathfrak{m})$ have the improved upper bound and further we obtain a refinement of these integral ideals in terms of prime powers. In particular, this enable us to deduce the bound for Fourier coefficients of elliptic cusp forms beyond the `typical size'. Moreover, we prove further improvements of the bound under the assumption of Littlewood's conjecture. Finally, We study a lower bound for the Fourier coefficients at prime powers provided the corresponding Hecke eigen angle is badly approximable.
