Rational Points and Analytic Number Theory
نویسنده
چکیده
There are a number of distinct ways in which analytic number theory can be used to provide information about rational points on algebraic varieties. Conversely, there are also a number of ways in which hoped-for results on the distribution of rational points could be used in classical problems from analytic number theory. Thus the analytic number theorist hopes not only to contribute to the theory of rational points, but also to get something back in return!
منابع مشابه
Arithmetic Elliptic Curves in General Position
We combine various well-known techniques from the theory of heights, the theory of “noncritical Belyi maps”, and classical analytic number theory to conclude that the “ABC Conjecture”, or, equivalently, the so-called “Effective Mordell Conjecture”, holds for arbitrary rational points of the projective line minus three points if and only if it holds for rational points which are in “sufficiently...
متن کاملExplicit methods for rational points on curves
One of the “big problems” of number theory is to understand the set of rational points on a variety, or equivalently, the rational solutions to a system of polynomial equations. Despite thousands of years of research, we are very far from having a general method for solving all such problems. There is even some evidence that deciding the existence of a rational solution is an undecidable proble...
متن کاملModuli Theory and Arithmetic of Algebraic Varieties
This paper surveys a few applications of algebro-geometric moduli theory to issues concerning the distribution of rational points in algebraic varieties. A few well known arithmetic problems with their expected answers (the socalled “diophantine conjectures”) are introduced in section 2, explaining their connection with a circle of ideas, whose goal is to find a unifying theme in analytic, diff...
متن کاملRational points near manifolds and metric Diophantine approximation
This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarńık type theorems for submanifolds of Rn. These problems have attracted a lot of interest since Kleinbock and Margulis proved a related conjecture of Alan Baker and V.G. Sprindžuk. They have been settled for planar curves but remain open in higher dimensions. In this...
متن کاملNevanlinna Theory and Diophantine Approximation
As observed originally by C. Osgood, certain statements in value distribution theory bear a strong resemblance to certain statements in diophantine approximation, and their corollaries for holomorphic curves likewise resemble statements for integral and rational points on algebraic varieties. For example, if X is a compact Riemann surface of genus > 1, then there are no non-constant holomorphic...
متن کامل