Shimara Cerresponchence of Hilbort modular forms

设K是全实代数域。我们考虑K上的Hilbert-Eisenstein级数，首先设K是实的二次级数。然后，我们发现了将Hilbert-Eisenstein级数限制在对角线上所得到的椭圆模形式与theta级数和Eisenstein级数的乘积的半整重模形式之间的一些关系。由此证明，所有重量至少为5/2的模形式都可以提升到Shimura意义下的整权模形式(请注意，这只对尖形形式是已知的)。作为应用，我们可以通过计算模形式的傅里叶系数以及一些算术函数之间的关系来得到Dirichlet L-函数的特殊值的公式。设F是K的二次扩张的全实代数域，我们考虑了将F上的Hilbert-Eisenstein级数限制到K上得到的K上的Hilbert模形式，并研究了它们在全代数数域K的数论上的作用.当K的Hilbert模形式的分次环的结构已知时，这种方法同样适用于椭圆模形式.我们发现了K上Dede型Zeta函数的特定值或正二次型表示的个数的几个公式.我们还研究了K上Hilbert模形式的Shimura对应.从而证明了(theta级数)x Eisenstein级数形式的Hilbert模形式可以提升为整权的Hilbert模形式.在椭圆的情况下，由此推论，所有模形式的半整数权可以被提升，然而在这种情况下，我们需要进一步的研究。

Let K be a totally real algebraic number field. We consider the Hilbert-Eisenstein series on K.At first let .K be real and quadratic. Then we discovered some relation between the elliptic modular forms obtained by restricting the Hilbert-Eisenstein series to the diagonal, and modular forms of half integral weight which are products of theta series and Eisensteinseries. By this it is shown that all modular forms of weight at least 5/2can be lifted to modular forms of integral weight in Shimura's sense (notethat this has been known only for cusp forms). As the application, we canobtain formulas for special values of the Dirichlet L-functions by computing Fourier coefficients of the modular forms, as well as relations between some arithmetic functions.Secondly let K be a general totally real algebraic number field. Let F bea totally real algebraic number filed which is a quadratic extension of K.We consider the Hilbert modular forms on K obtained by restricting the Hilbert-Eisenstein series on F to K, and investigate how they work on number theory of a totally algebraic number fields K.When the structure of graded ring of Hilbert modular forms of K is known, this method works well as in the case of elliptic modular forms. We discovered several formulas for special values for Dede kind zeta functions or the number of representations of positive quadratic forms over K.We investigate also the Shimura correspondence of Hilbert modular forms over K.As a result it is prove that the Hilbertmodular form in the form of (theta series) x Eisenstein series, can be lifted to a Hilbert modular form of integral weight. In the elliptic case it follows from this that all modular form of half integral weight can be lifted, however in this case we need further investigation.