Elliptic Curves and L-functions

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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.**