Heegner点和Eisenstein级数的算术

(A) Arithmetics of Heegner points over function fields and their relation to the number field case. Arithmetics of Heegner points on elliptic curves over number fields, were first studied by the constructive solution of the congruence number problem by Heegner. After then, people made a lot of important progress by using this on the research of the millionium problem, the B-SD conjecture and the congruent number problem. We hope to study the arithmetics over function fields, and its relation to the case over number fields. At present, this is a brand new topic in the world, and especially, Yun-Zhang’s recent work push such topic that we initiated to the climax of the whole arithmetic geometry..(B) Fourier coefficients formulae of Eisenstein series of general reductive groups. In earlier days, Shimura etc. got some results on classical groups, and the special exceptional group G2. We hope to use theta theory in the computations of the coefficients of the Eisenstein series, to seek the thetaversion proof for the known cases, to generalize to other cases, and also to consider the arithmetic application.

(A) 函数域上的Heegner 点的算术，以及它们和数域情形的关系。数域上的椭圆曲线上的.Heegner 点算术的研究，起源于K. Heegner 早年构造性地求解同余数问题。之后人们将其用于千禧问题BSD 猜想以及同余数问题的研究并取得了许多丰富的结果。我们希望考察函数域上的Heegner 点的算术，以及它们和数域情形的关系。特别是Heegner 点和L-函数的关系,它们在二次纽族里的变化行为,非平凡性的判断等。当前这是一个崭新的课题。. (B) 一般群的Eisenstein 级数的Fourier 系数公式，及其算术应用。早年， Shimura 等人对典型群以及特殊例外群G2 情形取得一些结果。我们希望通过将Theta 对应理论，运用于Eisenstein 级数的系数的计算的思想，寻求已知情形的公式的theta 证明，推广到更一般的情况，并考察其算术运用。

面向数论研究中的前沿领域，围绕Heegner点的算术、一般群的Eisenstein级数这两个前沿课题，在Langlands纲领的重数问题、Siegel-Weil公式的推广、椭圆曲线的二次扭族性质、Leopoldt 猜想等方面取得了突破。