论文标题
Siegel尖端2的傅立叶系数和Hecke特征值2
On Fourier coefficients and Hecke eigenvalues of Siegel cusp forms of degree 2
论文作者
论文摘要
我们研究了标量价值的Siegel cusp $ 2，重量$ K $和级别$ n $的傅立叶系数和Hecke特征值的一些关键分析性能。首先，假设$ f $是不属于锡托 - 川川类型的hecke特征类型，我们证明了在其Hecke Eigenvalue负面的最小素数的$ k $中的改进。其次，我们表明，在算术进展中，素数的hecke特征值$ f $中有无限的符号变化。第三，我们表明，在任何``radial''序列中都有无限多个积极和无限的负傅立叶系数，包括固定基本矩阵的主要倍数。最后，我们考虑$ f $是saito-saito-saito-kurokawa type，在这种情况下，我们认为（本质上是敏锐的）by（本质上是敏锐的）by $ | a（t） \ det t \ big）^{\ frac {k-1} {2}+ε} $对于$ f $的傅立叶系数时，只要$ \ gcd（$ \ gcd（4 \ det（t），n）$是SquareFree，确认了Das和Kohnen的猜想（在这种情况下是猜测$ n = 1 $）。
We investigate some key analytic properties of Fourier coefficients and Hecke eigenvalues attached to scalar-valued Siegel cusp forms $F$ of degree 2, weight $k$ and level $N$. First, assuming that $F$ is a Hecke eigenform that is not of Saito-Kurokawa type, we prove an improved bound in the $k$-aspect for the smallest prime at which its Hecke eigenvalue is negative. Secondly, we show that there are infinitely many sign changes among the Hecke eigenvalues of $F$ at primes lying in an arithmetic progression. Third, we show that there are infinitely many positive as well as infinitely many negative Fourier coefficients in any ``radial" sequence comprising of prime multiples of a fixed fundamental matrix. Finally we consider the case when $F$ is of Saito--Kurokawa type, and in this case we prove the (essentially sharp) bound $| a(T) | ~\ll_{F, ε}~ \big( \det T \big)^{\frac{k-1}{2}+ε}$ for the Fourier coefficients of $F$ whenever $\gcd(4 \det(T), N)$ is squarefree, confirming a conjecture made (in the case $N=1$) by Das and Kohnen.
