Analytic problems for L-functions and elliptic curves

本研究主要围绕L函数及其应用展开。L函数在现代数论中占有重要地位。近年来，L函数在几何性质和几何味道问题中的应用引起了人们极大的兴趣。在这项研究中，我想首先通过考虑椭圆曲线理论中的一些分析问题来继续我对L函数及其应用的研究。短期目标包括研究定义在有理数上的椭圆曲线(以及更一般的交换变种)的除域上的完全分裂素数的分布，以及研究椭圆曲线的约化模p上具有固定点数的素数p的个数。其次，本研究探讨了L函数的零点与某些算术函数的行为之间的密切联系。例如，作为一个短期目标，在关于Riemann Zeta函数零点的某些标准假设下，我们希望为Merten猜想成立的正数集的对数密度建立一个数值。第三，本研究旨在通过对数论的各种L功能的一般性研究，来更好地理解它们。这里的一个样本问题是研究一般L函数的幂阶矩的精确上界。这样的界限将给我们对L函数的行为有重要的认识。这项研究的成果将丰富我们在数论的几个重要领域的知识，并将有助于为该领域的进一步研究铺平道路。

This proposed research is centered around L-functions and their applications. L-functions play an important role in modern number theory. In recent years, there has been tremendous interest in the applications of L-functions to studying problems of geometric nature and flavor. In this research I would like to first continue my investigations on L-functions and their applications by considering some analytical problems arising from the theory of elliptic curves. The short term goals include studying the distribution of totally split primes in the division fields of an elliptic curve (and more generally an abelian variety) defined over rationals and studying the number of primes p with fixed number of points on the reduction mod p of an elliptic curve. Secondly this research explores the intimate connections between the zeros of L-functions and the behavior of certain arithmetic functions. For example as a short term goal, under certain standard assumptions on the zeros of the Riemann zeta function, we would like to establish a numerical value for the logarithmic density of the set of positive numbers for which the Mertens conjecture is true. Thirdly this research aims at understanding more about the various L-functions of number theory by studying them in a general setting. A sample problem here is studying sharp upper bounds for the power moments of general L-functions. Such bounds will give us significant insights on the behavior of L-functions. The outcome of this research will enrich our knowledge in several important areas of number theory and will help to pave the way for further studies in the field.