Hyperbolic and Diophantine Analysis
نویسنده
چکیده
In this survey we consider Kobayashi hyperbolicity, in which there is an interplay between five notions: analytic notions of distance and measure; complex analytic notions; differential geometric notions of curvature (Chern and Ricci form); algebraic notions of "general type" (pseudo ampleness); arithmetic notions of rational points (existence of sections). I am especially interested in the relations of the first four notions with diophantine geometry, which historically has intermingled with complex differential geometry. One of the main points of this survey is to arrive at a certain number of conjectures in an attempt to describe at least some of these relations coherently. Throughout this article, by an algebraic set we mean the set of zeros of a finite number of homogeneous polynomials
منابع مشابه
Self-similar fractals and arithmetic dynamics
The concept of self-similarity on subsets of algebraic varieties is defined by considering algebraic endomorphisms of the variety as `similarity' maps. Self-similar fractals are subsets of algebraic varieties which can be written as a finite and disjoint union of `similar' copies. Fractals provide a framework in which, one can unite some results and conjectures in Diophantine g...
متن کاملNevanlinna Theory and Rational Points
S. Lang [L] conjectured in 1974 that a hyperbolic algebraic variety defined over a number field has only finitely many rational points, and its analogue over function fields. We discuss the Nevanlinna-Cartan theory over function fields of arbitrary dimension and apply it for Diophantine property of hyperbolic projective hypersurfaces (homogeneous Diophantine equations) constructed by Masuda-Nog...
متن کاملDiophantine Approximation and the Geometry of Limit Sets in Gromov Hyperbolic Metric Spaces
In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes what has until now been an ad hoc collection of results by many authors. In addition to providing much greater generality than any prior work, our results also give new insight into the nature of the connection between Dio...
متن کاملBounded Geodesics of Riemann Surfaces and Hyperbolic Manifolds
We study the set of bounded geodesies of hyperbolic manifolds. For general Riemann surfaces and for hyperbolic manifolds with some finiteness assumption on their geometry we determine its Hausdorff dimension. Some applications to diophantine approximation are included.
متن کاملGeodesic Cusp Excursions and Metric Diophantine Approximation
We derive several results that describe the rate at which a generic geodesic makes excursions into and out of a cusp on a finite area hyperbolic surface and relate them to approximation with respect to the orbit of infinity for an associated Fuchsian group. This provides proofs of some well known theorems from metric diophantine approximation in the context of Fuchsian groups. It also gives new...
متن کامل