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.
This project is in the area of diophantine geometry. This is the study of systems of polynomial equations, in which the variables are only allowed to take on values in the whole numbers, or in the rational numbers. The specific work contained in the project grew out of the observation that many theorems in diophantine geometry share a strong resemblance with theorems in Nevanlinna theory, a subfield of complex analysis. Primarily, the project sought to leverage an inequality in Nevanlinna theory proved by McQuillan to make advances in diophantine geometry. McQuillan had already partly extended this result to the function field case, using the theory of algebraic stacks, which many mathematicians find difficult and abstract. A Canadian mathematician, Xi Chen, developed a version of McQuillan's proof that used the more readily accessible tools of algebraic geometry. Vojta found some errors in that proof, but was able to help Chen fix them. Ultimately, though, this work led to the conclusion that a direct attempt to translate McQuillan's work into number theory would not succeed without a truly fundamental breakthrough. Instead, Vojta looked more closely at the Thue-Siegel method, which forms a basis for many proofs in diophantine geometry. He noted that the method works only on closed subvarieties of semiabelian varieties. In addition, he showed in the case of Faltings' proof of approximation by rational points to an ample divisor in an abelian variety, that the proof can be rearranged so that it used the group law on a larger semiabelian variety in essentially the same way as Faltings' proof for rational points on closed subvarieties of abelian varieties. Vojta calls this "strict Thue's method." Vojta also took up work on lemmas of Dyson and Roth, which play a key role in proofs involving the Thue-Siegel method. He extended his early work on Dyson's lemma to the situation involving a product of an arbitrary (finite) number of smooth projective curves of arbitrary genus, but involving vanishing at only one point. He also made some progress on extending this work to give a sharper variant of Roth's lemma. Based on the known fact that McQuillan's original inequality comes so naturally out of the methods of Nevanlinna theory, it is anticipated that its counterpart in number theory should be provable using the Thue-Siegel method. The grant also partially supported some graduate students at Berkeley, aiding our nation's efforts in improving STEM education.
