论文标题
关于Siegel尖尖的基本傅立叶系数2
On fundamental Fourier coefficients of Siegel cusp forms of degree 2
论文作者
论文摘要
令$ f $为2度的Siegel尖缘形式，甚至是重量$ k \ geq 2 $和奇数Squarefree级别$ n $。我们对基本矩阵$ s $ $ a（f，s）$ a $ f $ $ a（f，s）$ s $ s $ S $（即$ -4 det（s）$等于基本歧视的$ -4 det）进行了详细研究。我们证明，作为$ s $的基本矩阵的等价类别不同，$ det（s）\ asymp x $，序列$ a（f，s）$具有至少$ x^{1-ε} $符号变化，并且至少需要$ x^{1-ε} $“大价值”。此外，假设普遍的Riemann假设以及精致的Gan-Gross-- PRASAD猜想，我们证明了绑定的$ | a（f，s）| \ ll_ {f，ε} \ frac {\ det（s）^{\ frac {k} 2 - \ frac {1} {2}}}} {（\ log | \ det（s）|）
Let $F$ be a Siegel cusp form of degree 2, even weight $k \geq 2$ and odd squarefree level $N$. We undertake a detailed study of the analytic properties of Fourier coefficients $a(F,S)$ of $F$ at fundamental matrices $S$ (i.e., with $-4 det(S)$ equal to a fundamental discriminant). We prove that as $S$ varies along the equivalence classes of fundamental matrices with $det(S) \asymp X$, the sequence $a(F,S)$ has at least $X^{1-ε}$ sign changes, and takes at least $X^{1-ε}$ "large values". Furthermore, assuming the Generalized Riemann Hypothesis as well as the refined Gan--Gross--Prasad conjecture, we prove the bound $|a(F,S)| \ll_{F, ε} \frac{\det(S)^{\frac{k}2 - \frac{1}{2}}}{ (\log |\det(S)|)^{\frac18 - ε}}$ for fundamental matrices $S$.
