Application of theory of elliptic curves to algebraic topology

在这项研究中，我们发展了与泛椭圆曲线的形式群和椭圆上同调有关的群拟阵和Hopf-代数体的理论。我们研究了分次代数范畴中的模模层和模层理论。在将数论应用于代数拓扑学中，我们研究了泛椭圆曲线模型、局部域上的自同构表示和局部对称空间的上同调群。通过Weil-Taniyama猜想确定了小导体的椭圆曲线模型和与椭圆曲线深度相连的模曲线的模型，得到了非轴对称局部域上尖端非分支化的单代数列的特征标公式和局部对称空间上同调群的维数公式。在这些结果的基础上，我们研究了泛椭圆曲线的形式群与自同构表示的形式群之间的关系。此外，我们还利用在微分方程组和实分析领域中的新结果研究了自同构形式。

In the research, we developed the theory of groupoid schemes and Hopf-algebroid related to formal groups of universal elliptic curves and elliptic cohomology. We studied the theory of schemes and sheaves of modules in the category of graded algebras. In application of number theory to algebraic topology, we studied models of universal elliptic curves, automorphic representations over local fields and cohomology groups of locally symmetric spaces. We determined models of elliptic curves of small conductor and that of modular curves deeply connected to elliptic curves through Weil-Taniyama conjecture, We obtained a character formula of cuspidal unramified series of simple algebras over non-axchimedean local fields and a dimension formula of cohomology groups of locally symmetric spaces. Based on those results we studied relation between formal groups of universal elliptic curves and those of automorphic representations. Further we studied automorphic forms using the results newly obtained in the field of differential equations and real analysis.