Hilbert, modular functions and quadratic forms

设K是一个全实数代数域。与系数为K的正二次型相关的级数是希尔伯特模形式。我们考虑二次型的变量数很少的情况。然后，为了研究二次形式表示的个数，或者二次形式表示的个数与Dedekind l函数的特殊值之间的关系，我们需要研究低权值的Hilbert模形式的空间。主要处理K为实二次元的情况。实现了Hilbert模曲面上可约轨迹的描述。设O_K是K的最大阶数，而O_K是不同的。设λ， η为O_K的z基，且Nm(λη)<0， Nm表示范数映射。这样的λ η只有有限个，直到单位相乘。称U为（λ, η）模单位群的类的完全代表。让< <数值公式> >。那么U与Hilbert模曲面上与Γ_<OK>相关的可约轨迹集之间就存在自然的一一对应关系。V.A.Gritsenko和V.V.Nikulin证明了在2次的西格尔模情况下，级数被写成无穷积。从模理论出发，得到了与Γ_<OK>相关的Hilbert模曲面到二阶Siegel模变体的自然映射，称为模嵌入。Hilbert模情况的Theta级数是由Siegel情况的Theta级数通过与模嵌入相关的环同态得到的。由于与Γ_<OK>相关的Hilbert模曲面上的theta级数仅在可约轨迹处消失，我们得到了Hilbert模曲面上theta级数的无穷积表达式。进一步研究了权值1/2或3/2的希尔伯特模形式的维数的求取条件。然而，我们需要对此进行进一步的调查。

Let K be a totally real algebraic number field. The theta series associated with a positive quadratic form with coefficients in K is a Hilbert modular form. We consider the case that the number of the variables of the quadratic form is small. Then to investigate the numbers of representations by the quadratic form, or the relation between the numbers of the representations by the quadratic form and the special values of Dedekind L-function, we need to investigate the space of Hilbert modular forms of low weight.The case that K be real quadratic, is mainly treated. It is accomplished to describe reducible loci on the Hilbert modular surface. Let O_K be the maximal order of K, and let O_K be the different. Let λ, η be the Z-base of O_K with Nm(λη)<0, Nm denoting the norm map. Then there are only a finite number of such λ, η up to multiplications by units. Call U, the complete representatives of the classes of (λ, η) modulo the unit group. Let<<numerical formula>>.Then there is the natural one to one correspondence between U and the set of reducible loci on the Hilbert modular surface associated with Γ_<OK>. V.A.Gritsenko and V.V.Nikulin showed that in the Siegel modular case of degree 2, the theta series is written as a infinite product. From moduli theory, there is the natural map of the Hilbert modular surfaces associated with Γ_<OK> into Siegel modular variety of degree two, which is called a modular embedding. Theta series of Hilbert modular case are obtained from theta series of Siegel case via the ring homomorphism associated with the modular embedding. Since theta series on the Hilbert modular surface associated with Γ_<OK> vanishes only at reducible loci, we obtain the infinite product expression of theta series on the Hilbert modular surface. Furthermore we study the condition under which the dimension of the Hilbert modular forms of weight 1/2 or 3/2 be obtained. However we need further investigation about this.