本项目将着重于整体域（数域和整体函数域）椭圆曲线算术性质的研究。我们将研究数论最新方法在函数域椭圆曲线上的类比，特别是探求Heegner点理论，Gross-Zagier公式在函数域上的类比，并由所得到的结果反过来启发数域椭圆曲线算术性质的研究。我们将研究p进伽罗瓦表示理论在椭圆曲线算术上的应用，特别是关于p进L函数的构造和Iwasawa理论，欧拉系理论，p进Gross-Zagier公式以及一些情形的p进BSD猜想。我们还将探索Scholze理论在椭圆曲线算术理论中的应用。

This project is a study on the arithmetic of elliptic curves over global fields (i.e. number fields and global function fields). We shall study analogue of the latest number fields methods on function fields, especially the analogue of Heegner point theory and Gross-Zagier formula on function fields, and then use the results obtained to enlighten the study of arithmetic properties of elliptic curves over number fields. We shall study the application of the theory of p-adic Galois representations on the arithmetic of elliptic curves. In particular, we shall study the construction of p-adic L-functions and Iwasawa theory, Euler system theory, p-adic Gross-Zagier formula and p-adic BSD Conjetcures for some family of elliptic curves. We shall also explore the application of Scholze theory on the arithmetic of elliptic curves.