论文标题
在Hermitian Eisenstein系列第二级
On Hermitian Eisenstein series of degree 2
论文作者
论文摘要
我们考虑了Hermitian Eisenstein系列$ e^{（\ Mathbb {k}）} _ k $ $ 2 $ 2 $和重量$ k $与虚构的Quadratic数字$ \ MATHBB {K k} $相关，并确定$ \ \ Mathbb {k} $对其Arithmetic and the Fruenditiations的影响。我们发现他们满足了身份$ e^{{（\ Mathbb {k}）}}^2} _4 = e^{{（（\ Mathbb {k}}} _ 8 $，这是Siegel Modular Modular of度数$ 2 $，并且仅在$ \ Mathbb {K} k}} = { （\ sqrt {-3}）$。作为一个应用程序，我们表明Eisenstein系列$ e^{（\ Mathbb {k}）} _ k $，$ k = 4,6,8,10,12 $在$ \ Mathbb {k} \ neq \ neq \ Mathbb {q}（q}（q}（q}（q}（Q}（Q}（\ sqrt））时， Siegel和Hermitian对Siegel半空间的限制之间的差异是Maass空间中的尖端形式，其重量足够大。但是，当重量固定时，我们会看到它趋于$ 0 $，因为判别倾向于$ - \ infty $。最后，我们表明这些形式在Maass Spezialschar中生成了尖峰形式的空间作为Hecke代数上的模块，因为$ \ Mathbb {k} $在虚构的Quadratic数字字段上有所不同。
We consider the Hermitian Eisenstein series $E^{(\mathbb{K})}_k$ of degree $2$ and weight $k$ associated with an imaginary-quadratic number field $\mathbb{K}$ and determine the influence of $\mathbb{K}$ on the arithmetic and the growth of its Fourier coefficients. We find that they satisfy the identity $E^{{(\mathbb{K})}^2}_4 = E^{{(\mathbb{K})}}_8$, which is well-known for Siegel modular forms of degree $2$, if and only if $\mathbb{K} = \mathbb{Q} (\sqrt{-3})$. As an application, we show that the Eisenstein series $E^{(\mathbb{K})}_k$, $k=4,6,8,10,12$ are algebraically independent whenever $\mathbb{K}\neq \mathbb{Q}(\sqrt{-3})$. The difference between the Siegel and the restriction of the Hermitian to the Siegel half-space is a cusp form in the Maass space that does not vanish identically for sufficiently large weight; however, when the weight is fixed, we will see that it tends to $0$ as the discriminant tends to $-\infty$. Finally, we show that these forms generate the space of cusp forms in the Maass Spezialschar as a module over the Hecke algebra as $\mathbb{K}$ varies over imaginary-quadratic number fields.
