Study on elliptic cohomology theory

椭圆上同调理论研究

• 批准号: 08404002
• 负责人: NISHIDA Goro
• 金额: $ 7.87万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 1996
• 资助国家: 日本
• 起止时间: 1996 至 1997
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08404002/
• 关键词: Elliptic cohomology; Homotopy theory; Elliptic curve; Modular-form; Formal group low; Hopf algebra; Classifying space; Elliptic genera; ホモトピー論; 楕円曲線; ホップ代数; 楕円種数

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.

Landweber用一个与椭圆曲线相关的形式群low定义了椭圆上同伦理论，用同伦理论来表示椭圆属。本课题组通过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的关系。作为一个显式的结果，我们证明了n阶模形式的环可以用n阶循环群的分类空间的椭圆上同调来标识。当n趋于无穷时，这些椭圆上同调环近似于形式Hopf。表示椭圆上同调理论的形式群低的代数。这意味着可以由高阶模形式的伽罗瓦理论构造椭圆上同调理论。作为其他相关的结果，我们证明了作用于n阶循环群的分类空间的m倍积的上同群的一般线性群的表示与Lubin-Tate的高度n形式群low的乘n序列的分裂场的伽罗瓦表示有密切的关系。研究一般线性群的表示与Eichler-Shirura上同调的关系是一个很有意义的问题。

项目成果

M.Maruyama: "Construction of moduli spaces of stable sheave via Simpson's idea" Proc.of Taniguchi Symp.“Moduli of Vector Bundles". 147-188 (1996)

M.Maruyama：“通过辛普森的想法构造稳定滑轮的模空间”，Taniguchi Symp 的“矢量束模量”147-188 (1996)。

M.Maruyama: "Construction of moduli spaces of stable sheave via Simpson's idea" Proc.of Taniguchi Symp."Moduli of Vector Bundles". 147-188 (1996)

M.Maruyama：“通过辛普森的想法构造稳定滑轮的模空间”Proc.of Taniguchi Symp.“矢量束的模量”。

S.Morimoto, G.Nishida: "Elliptic cohomology of classifying spaces of cyclic groups ans higher level modular forms" J.Math.Kyoto Univ.37. 701-715 (1998)

S.Morimoto、G.Nishida：“循环群分类空间和高级模形式的椭圆上同调”J.Math.Kyoto Univ.37。

G.Nishida: "A duality theorem in Hopf algebras and its application to Morava K theory of BZ/p" J.Math.Kyoto Univ.36. 1-8 (1997)

G.Nishida：“Hopf 代数中的对偶定理及其在 BZ/p 的 Morava K 理论中的应用”J.Math.Kyoto Univ.36。

H.Yoshida: "On absolute CM-periods" Lecture Note Series, RIMS,Kyoto Univ.965. 87-133 (1996)

H.Yoshida：“论绝对 CM 周期”讲义系列，RIMS，京都大学 965。