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-函数在现代数论中起着重要的作用。近年来，L-函数在研究几何性质和味问题中的应用引起了人们极大的兴趣。在这项研究中，我想首先继续我的调查L-函数及其应用，考虑一些分析问题所产生的椭圆曲线理论。 短期目标包括研究在有理数上定义的椭圆曲线（更一般地说是阿贝尔簇）的划分域中完全分裂素数的分布，以及研究在椭圆曲线的约化模p上具有固定点数的素数p的数量。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，在黎曼zeta函数的零点的某些标准假设下，我们想建立一个数值的对数密度的一组正数的Mertens猜想是真的。 第三，本研究的目的是通过在一般环境下研究数论中的各种L-函数来更好地理解它们。这里的一个示例问题是研究一般L-函数的幂矩的上界。这样的界限将给我们提供关于L-函数行为的重要见解。这项研究的成果将丰富我们在数论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.