Diophantine approximation and Nevanlinna theory

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

The analogy between number theory and Nevanlinna theory has led to much interplay between the two fields in both directions, but at its most fundamental level is not well understood. In particular, the use of the derivative of a holomorphic function in Nevanlinna theory stands out as something with no known counterpart in number theory. The recent "tautological inequality" of M. McQuillan, however, is of a form that can be translated into number theory, leading to a conjecture that the present project will investigate. The project includes work on trying to prove this conjecture, starting with special cases stemming from the Subspace Theorem of W. M. Schmidt, and from Faltings' work on closed subvarieties of abelian varieties. In addition, it will further study the extent to which McQuillan's inequality can serve as a gateway to the main theorems of Nevanlinna theory. As a separate but related project, the Principal Investigator will also continue work on completed invariant jet spaces in the arithmetical context. The Thue-Siegel-Roth method in number theory is a method for showing that certain types of diophantine equations have only finitely many solutions, or at least showing that their families of solutions obey additional equations. It made its debut 100 years ago this year, but it has been gaining momentum in the last 20 years, due in part to similarities with Nevanlinna theory. The latter is a part of complex analysis, encompassing methods for showing that meromorphic functions having certain properties do not exist, or more generally that in certain cases a non-constant holomorphic function from the complex plane to a complex algebraic manifold must satisfy additional equations. The similarities between these two fields have benefited both areas of mathematics, allowing ideas, conjectures, and methods to be carried over from one area to the other, in both directions. However, the fundamental reasons for these similarities are not at all understood at the present time. In particular, the derivative -- specifically the "lemma on the logarithmic derivative" -- plays a central role in Nevanlinna theory, but has no known counterpart in number theory. Based on a recent inequality of McQuillan in Nevanlinna theory, however, the proposer has a conjectural counterpart for this lemma in number theory. This grant will support work on this conjecture and its possible ramifications.