Hilbert, modular functions and quadratic forms

设K是全实代数域。与系数在K中的正二次型相关的theta级数是Hilbert模形式。我们考虑了二次型变量个数较少的情况。为了研究二次型表示的个数，或者二次型表示的个数与Dedekind型L函数的特定值之间的关系，我们需要研究低权重的Hilbert模型空间，主要讨论了K是实二次型的情况。实现了对Hilbert模曲面上可约轨迹的刻画。设O_K是K的最大阶，O_K是K的最大阶。设λ，η是O_K的Z-基，N_m(λη)&lt；0，N_m表示范数映射。那么这样的λ，η只有有限数量，直到单位乘法。称U为模为单位群的(λ，η)类的完全代表。令&lt；&lt；数值公式&gt；，则U与与Γ；&lt；OK&gt；有关的希尔伯特模曲面上的可约轨迹集之间存在自然的一一对应。V.A.Gritsenko和V.V.Niklin证明了在二次Siegel模情形下，theta级数表示为无穷多个积。从模论出发，得到了与Γ_&lt；OK&gt；相联系的Hilbert模曲面到二次Siegel模簇的自然映射，称为模嵌入。通过与模嵌入相关的环同态，由Siegel壳的theta级数得到Hilbert模壳的theta级数.由于与Γ_&lt；OK&gt；相关的Hilbert模曲面上的theta级数仅在可约轨迹处消失，因此我们得到了Hilbert模曲面上的theta级数的无限乘积表达式。此外，我们还研究了权为1/2或3/2的Hilbert模的维数的条件。然而，我们还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.