Study on elliptic cohomology theory

椭圆上同调理论是Landweber用椭圆曲线的形式群来表示同伦理论中的椭圆亏格而定义的。本课题组利用椭圆上同调理论研究了同伦理论与椭圆曲线和模形式理论之间的关系。作为一个明确的结果，我们证明了n阶模形式环可以与n阶循环群的分类空间的椭圆上同调相一致。当n趋于无穷大时，这些椭圆上同调环逼近形式Hopf代数，它代表椭圆上同调理论的形式群。这意味着，从高阶模形式的伽罗瓦理论中构造椭圆上同调理论是可能的。作为其它相关结果，我们证明了作用于n阶循环群的分类空间的m次积的上同调群的一般线性群的表示与Lubin-Tate的高n形式群low的n乘序列的分裂域的Galois表示之间有着密切的联系。研究一般线性群的表示与Eichler-Shirura上同调的关系是一个有趣的课题。

The elliptic cohomology theory was defined by Landweber using a formal group low associated with elliptic curves to express the elliptic genera in terms of homotopy theory. Our research group studied relations between the homotopy theory and the theory of elliptic curves and modular forms via the elliptic cohomology theory. As an explicit result, we showed that the ring of level n modular forms can be identified with the elliptic cohomology of the classifying space of the cyclic group of order n. When n tends to infinity, these elliptic cohomology rings approximates the formal Hopf.algebra which represents the formal group low of the elliptic cohomology theory. This means that it may be possible to construct the elliptic cohomology theory from the Galois theory of higher level modular forms. As other related result, we showed that there is a close relation between the representation of general linear group acting on the cohomology group of m-times product of the classifying spaces of the cyclic group of order n and the Galois representation of a splitting field of the multiplication-by-n sequence of the height n formal group low of Lubin-Tate. As a fortune program, it is interesting to study the relation of the representation of general linear group to the Eichler-Shirura cohomology.