申请人计划构造出一类被称为F虚拟GL(2)型阿贝尔簇的具体实例。它们具有较大的自同态代数，可以导出GL(2)型的Galois表示，它们的isogeny类定义在给定数域F上。申请人将构造并研究它们的参数空间的性质，构造参数空间上的有理点，运用Gross-Zagier-张公式进行计算，在这些实例上检测BSD猜想可行性。申请人还将研究(τ,b)理论，即研究L函数、周期积分和Endoscopy对应之间的关系。Endoscopy对应可以看成Theta对应的一种推广。周期积分的消失与否能给出相关L函数的极点的位置的信息，从而得到尖自守表示的Arthur参数中(τ,b)形式因子的存在性。申请人还将考虑Rallis内积公式的算术版本，研究L函数在中心处导数和算术Theta提升的Beilinson-Bloch高度配对的联系。

We plan to construct concrete F-virtual Abelian varieties of GL(2)-type. They are Abelian varieties with big endomorphism algebra which induce Galois representations of GL(2)-type and their isogeny class is defined over the given number field F. We will construct and study properties of their parameter spaces, construct rational points on the parameter spaces and apply Gross-Zagier-Zhang formula. We hope that our work will furnish examples against which the BSD conjecture may be tested. We also plan to study (τ,b)-theory which concerns the relation among L-functions, period integrals and Endoscopy correspondence. Endoscopy correspondence can be viewed as a generalisation of theta correspondence. The non-vanishing or vanishing of period integrals can give information on the locations of poles of related L-functions and thus detect the existence of factors of the form (τ,b) in Arthur parameters of cuspidal automorphic representations. We plan also to consider the arithmetic Rallis inner product formula. We will study the relation between central derivative of L-functions and the Beilinson-Bloch height pairing of arithmetic theta lifts.