Elliptic Curves and L-functions

这个研究项目是在数论领域，这是数学的一部分，涉及整数的性质和整数系数方程的解。这样的方程称为丢番图方程。椭圆曲线是丢番图方程理论的基本研究对象之一。椭圆曲线的研究可以用代数、分析和几何的方法进行。在过去的十年中，我研究了几个问题，在这方面从分析的角度来看，并采用L-函数，这是基本的工具，在解析数论。 在这项研究中，我计划继续我在椭圆曲线和L-函数的调查。短期目标包括研究两条椭圆曲线的分布问题，研究与椭圆曲线上点的倍数的分母相关的椭圆分母序列的算术性质，找到短间隔内L函数系数和的显式界，深入理解经典的通过概率方法的数论误差项，建立了某些L-函数的特殊值的分布定理。 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 research program is in the field of number theory, which is the part of mathematics that deals with properties of integers and solutions of equations with integer coefficients. Such equations are called Diophantine equations. Elliptic curves are one of the fundamental objects of study in the theory of Diophantine equations. The study of elliptic curves is amenable to methods from algebra, analysis, and geometry. Over the past decade, I have studied several problems in this area from an analytic point of view and by employing L-functions which are fundamental tools in analytic number theory. In this research I plan to continue my investigations in elliptic curves and L-functions. The short term goals include studying distribution problems for two elliptic curves, investigating arithmetical properties of elliptic denominator sequences associated to the denominators of multiples of points on elliptic curves, finding explicit bounds for sums of coefficients of L-functions in short intervals, in depth understanding of the classical error terms of number theory via probabilistic methods, and establishing distribution theorems for special values of certain 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.