The PI will study several interrelated topics connected with the arithmetic of higher-dimensional varieties, including related questions in the value distribution theory of holomorphic curves. In one direction, he will study the problem of proving the finiteness, or more generally bounding the dimension, of the set of integral points on affine varieties having many components "at infinity". Results here will constitute progress towards proving the PI's conjectures for integral points on such varieties, which may be viewed as higher-dimensional versions of Siegel's theorem for integral points on affine curves. An additional novel aspect is the PI's plan to adapt effective methods in number theory to this higher-dimensional setting. Applications of these results to families of equations, modular varieties, and problems in arithmetic dynamics will be explored. Among the tools used will be the Schmidt subspace theorem from Diophantine approximation, a fundamental and deep tool in number theory. In fact, some of the techniques employed in the proofs of these results are expected to yield variations and improvements of the Schmidt subspace theorem itself, for instance to the setting of algebraic points of bounded degree. From the work of Vojta and others, it has been discovered that many statements in Diophantine approximation, when stated appropriately, bear a strong resemblance to statements in Nevanlinna theory, a branch of complex analysis. Through the dictionary between the two subjects developed by Vojta, it is frequently possible to take statements and proofs in one subject and develop analogous statements and proofs in the other subject. In this manner, the PI expects to prove analogous results in Nevanlinna theory, where Schmidt's theorem corresponds to Cartan's Second Main Theorem, and qualitatively, to prove results for holomorphic curves analogous to results for integral points.
The research proposed here revolves around one of the oldest and most fundamental problems in mathematics: understanding the set of solutions to a system of polynomial equations in rational numbers or integers. The proposed research would contribute substantially to understanding this difficult and basic problem by providing new results and techniques in several basic contexts. Moreover, the research has ramifications across other areas of mathematics. Most notably, it is expected to yield results in complex analysis, and to enrich our understanding of the deep links between complex analysis and number theory.
The PI studied several interrelated topics connected with the arithmetic of varieties. At a basic level, the research revolved around one of the oldest and most fundamental problems in mathematics: understanding the set of solutions to a system of polynomial equations in rational numbers or integers. The outcomes of the project included original research and advances in the understanding of Diophantine approximation, Diophantine geometry, arithmetic dynamics, value distribution theory, and p-adic Nevanlinna theory. This research has resulted in six published peer-reviewed papers, two peer-reviewed papers accepted for publication, and three papers under review: Wirsing-type inequalities (submitted) Integral points of bounded degree on affine curves (submitted) The Nagell-Ljunggren equation via Runge's method (submitted) (with Mike Bennett) On the p-adic Second Main Theorem, Proc. Amer. Math. Soc. (accepted) On the Schmidt Subspace Theorem for algebraic points, Duke Math J. (accepted) Linear forms in logarithms and integral points on higher-dimensional varieties, Algebra Number Theory 8 (2014), no. 3, 647–687. Siegel's Theorem and the Shafarevich Conjecture, J. Théor. Nombres Bordeaux 24 (2012), no. 3, p. 705-727. Pulling back torsion line bundles to ideal classes, Math. Res. Lett. 19 (2012), no. 6, 1171-1184 (with Jean Gillibert). Rational preimages in families of dynamical systems, Monatsh. Math. 168 (2012), no. 3-4, 473–501. Ideals of degree one contribute most of the height, Algebra & Number Theory 6 (2012), no. 6, 1223-1238 (with David McKinnon). The exceptional set in Vojta's conjecture for algebraic points of bounded degree, Proc. Amer. Math. Soc. 140 (2012), no. 7, 2267-2277. Broader impacts of the project include the co-supervision of two undergraduate research projects at Michigan State University (projects of Zishen Yang and Wang Wei), supervision of graduate students in the doctoral program, and the co-organization of two conferences: a conference at BIRS in Banff on Vojta’s conjectures (September 2014) and an AMS special session at UC Riverside on Diophantine geometry and Nevanlinna theory (November 2013).
