A Study on Jacobi Forms by a Method of Algebraic Geometry

我们计算了权为k的二次斜全纯Jacobi形式空间的维度。这是以前计算二次全纯Jacobi形式空间的维数的结果的推广。此外，我们还计算了在向量值情况下计算斜全纯Jacobi形式空间的维度所需的Riemann-Roch公式。这还不是一个维度公式，因为我们还没有证明消失定理。如果我们将T.Ibukiyama的结果应用到我们的结果中，我们得到了半整权Siegel模形式空间中称为正空间的子空间的维度公式(这是向量值情况下的猜想)。半整数权的Siegel模形式的空间是由这些空间的量纲公式的前人工作明确地确定的。加号空格也是e…更明确地确定(由于S·Hayashida的计算机计算)。Saito-Kurokawa的Siegel模形式的提升理论是由加空间和高次Jacobi形式空间的同构构造的。利用这个结果，完成了从二次到三次的提升理论。在20世纪70年代，S·G·下村发现了一次整权的模形式到半整权的模形式的对应，这就是今天的下村对应。这是一个自然的想法，尝试将这种对应的情况推广到更高的程度。但30年后，线索仍未找到。由于Shimura对应不是半整数权的模形式空间的满射，因此知道期望作为对应的像的正空间的维度是重要的。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。如果我们比较二次Siegel模空间和整权Siegel模空间的维度与Jacobi形式空间的维度，我们就可以确定期望与整权Siegel模空间同构的正空间的维度。众所周知，这些空间的尺寸彼此相等(由于T·Ibukiyama)。由此推测了二次情形下的Shimura对应的存在性。较少

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