代数型指数和及其应用

代数型指数和是现代数论与代数几何的重要研究对象，在孪生素数猜想、Riemann zeta函数的零点密度估计、GL(3) L函数的亚凸界等问题的重大进展中发挥了关键的作用，为代数几何应用于现代解析数论建立了重要的桥梁。本项目拟研究代数型指数和的解析理论及其在现代数论中的应用。首先，发展平均形式的van der Corput方法及混合型指数对理论，对脆数模的代数型指数和给出系统的上界估计；其次，基于van der Corput方法研究多项式表素数的Schinzel猜想以及GL(3)自守形式Fourier系数与Kloosterman和的均值估计等。代数型指数和及其应用方面的研究，体现了经典解析数论、代数几何及自守形式等不同分支的交叉融合，具有重要的理论意义与广阔的应用前景。

As important subjects in modern number theory and algebraic geometry, algebraic exponential sums have established important bridges for applications of the latter to the former. In particular, they have played key roles in the recent breakthroughs in the twin prime conjecture, zero density of Riemann zeta functions and subconvexity for GL(3) L-functions. This proposal is devoted to the analytic theory of algebraic exponential sums and its applications to modern number theory. Precisely, an averaged van der Corput method and hybrid exponential pairs will be developed, so that some systematic estimates for algebraic exponential sums with smooth moduli will be established. Based on the new van der Corput method, the Schinzel hypothesis for presenting primes by polynomials will be considered, as well as the mean values of Fourier coefficients of automorphic forms and Kloosterman sums for GL(3). Analytic number theory, algebraic geometry and automorphic forms are combined and interact with each other while studying modern analytic number theory using algebraic exponential sums, the systematic study of which will definitely prove important developments in theory and fruitful applications in practice.