On Fourier Coefficients and Hecke Eigenvalues of Siegel Cusp Forms of Degree 2
On Fourier Coefficients and Hecke Eigenvalues of Siegel Cusp Forms of Degree 2
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DOI：
10.1093/imrn/rnac316
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发表时间：
2022-07
影响因子：
1
通讯作者：
Biplab Paul;A. Saha
中科院分区：
文献类型：
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作者：
Biplab Paul;A. Saha
本文研究了阶数为2、权为k、水平为N的标值Siegel尖点形式F的Fourier系数和Hecke特征值的一些重要的分析性质。首先，假设$F$是一个Hecke本征形，是不是斋藤黑川型，我们证明了一个改进的界在$k$-方面的最小素数，在其Hecke本征值是负的。其次，我们表明，有无穷多个符号之间的变化的Hecke特征值的$F$在素数躺在一个等差数列。第三，我们表明，有无穷多个积极的，以及无穷多个负傅立叶系数在任何“径向”序列组成的素数倍的一个固定的基本矩阵。最后，我们考虑了当$F$是Saito-Kurokawa型时的情形，在这种情形下，我们证明了（本质上尖锐的）界$|a（T）|当$\gcd（4 \det（T），N）$无平方时，$F$的傅立叶系数的~\ll _{F，\big}（\det T \big）^{\frac {k-1}{2}+\big}$，证实了Das和Kohnen的一个猜想（在$N=1$的情况下）。
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 $| a(T) | ~\ll _{F, \epsilon }~ \big ( \det T \big )^{\frac {k-1}{2}+\epsilon }$ 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.