A Study on Jacobi Forms by a Method of Algebraic Geometry

我们计算了权为k且度为2的斜全纯Jacobi形式空间的维数。这是前一个结果的推广，其中我们计算了二次全纯Jacobi形式空间的维数。此外，我们计算了Riemann-Roch公式，该公式在向量值情况下也是计算斜全纯Jacobi型空间的维数所必需的。这还不是维数公式，因为我们还没有证明消失定理。但这是计算维数公式所必需的，也足以估计维数公式。Ibukiyama在此基础上，得到了半整权Siegel模形式空间中称为正空间的子空间的维数公式（这是向量值情形下的一个猜想）.半整权Siegel模形式的空间是由这些空间的维数公式的已有工作明确确定的。加空格也是e 关于我们 实验测定（由于S. Hayashida）。Saito-Kurokawa的Siegel模形式的提升理论是由加空间与上次Jacobi形式空间的同构构造的。在70年代，G.Shimura发现了在一次情形下整权模形式与半整权模形式的对应关系，这就是今天所称的Shimura对应。尝试把这种对应关系推广到更高次的情形是一个自然的想法。但30年过去了，线索一直没有找到。由于志村对应不是半整权模形式空间的满射，所以知道期望成为对应的图像的正空间的维数是很重要的。此外，研究向量值模形式的一般情况是绝对重要的，因为高次Shimura对应是向量值模形式之间的对应。如果我们比较Siegel模形式的空间的维度2度和积分重量和Jacobi形式的空间的维度假设消失定理，我们可以确定的加空间的维度，预计同构的Siegel模形式的空间的积分重量。并且已知这些空间的维数彼此相等（由于T.Ibukiyama）。由此证明了在二阶情形下志村对应的存在性。少

We computed the dimension of the spaces of skew-holomorphic Jacobi forms of weight k and degree two. This is an extension of the former result in which we computed the dimension of the spaces of holomorphic Jacobi forms of degree two. Moreover we computed the formula of Riemann-Roch which is needed to compute the dimension of the spaces of skew-holomorphic Jacobi forms also in the vector valued case. This is not a dimension formula yet, since we have not proved the vanishing theorem. But this is needed to compute the dimension formula and sufficient to estimate the dimension formula.If we apply the result of T. Ibukiyama to our result, we obtain the dimension formula for the subspaces called plus spaces in the spaces of Siegel modular forms of half integral weight (this is a conjecture in the vector valued case). The spaces of Siegel modular forms of half integral weight were explicitly determined by the former work of the dimension formula for these spaces. The plus spaces were also e … More xplicitly determined (due to a computer calculation by S. Hayashida). The lifting theory of Siegel modular forms of Saito-Kurokawa is constructed by an isomorphism of the plus space and the space of Jacobi forms of upper degree. By this result the lifting theory from degree two to degree three is completed.In the 1970's G.Shimura found a correspondence from modular forms of integral weight to modular forms of half integral weight in the case of degree one which is called Shimura correspondence today. It is a natural idea to try an extension this correspondence to the case of higher degree. But the clue have not been found after 30 years. Since the Shimura correspondence is not a surjection to the space of modular forms of half integral weight, it is important to know the dimension of the plus space which is expected to be the image of the correspondence. Moreover it is absolutely important to study the general case of vector valued modular forms because the Shimura correspondence of higher degree is a correspondence between vector valued modular forms. If we compare the dimension of the space of Siegel modular forms of degree two and integral weight and the dimension of the space of Jacobi forms assuming the vanishing theorem, we can determine the dimension of the plus pace which is expected to be isomorphic to the space of Siegel modular forms of integral weight. And it is known that the dimensions of these spaces are equal to each other (due to T.Ibukiyama). From this the existence of the Shimura correspondence in the case of degree two is conjectured. Less