代数簇的算术问题是算术代数几何的核心问题之一。素数分布一直是数论的核心问题之一，素数在群作用的轨道上的分布规律，同时也是代数簇的算术问题的深化。朗兰兹纲领将代数簇的算术问题与自守表示联系起来，使得代数簇具体算术问题的L-函数与某个自守表示的L-函数相同，这样就把代数簇的算术问题化成自守表示的问题。本项目计划深入研究自守表示、自守L-函数、以及BSD猜想，并把成果用于代数簇的算术问题，以及素数分布领域的Sarnak猜想。具体地说，本项目拟研究以下三个方面的问题：自守L-函数；BSD猜想与Heegner点的算术理论；Sarnak猜想。

The arithmetic of algebraic varieties is central in arithmetic algebraic geometry. The distribution of primes is also central in number theory, while the distribution of primes on orbits of group actions is a deeper problem in the arithmetic of algebraic varieties. The Langlands program links an arithmetic problem with an automorphic representation in such a way that the L-functions involved are the same, and hence the study of arithmetic problems is transferred to that of automorphic representations. This project will focus on automorphic representations, automorphic L-functions, the BSD conjecture, and their application to the arithmetic of algebraic varieties. Special attention will be paid to the following three aspects: automorphic L-functions; BSD conjecture and the arithmetic of Heegner points; Sarnak’s conjecture.