A panoramic overview of inter-universal Teichmuller theory (Algebraic Number Theory and Related Topics 2012)

Abstract

Inter-universal Teichmüller theory may be described as a sort of arithmetic version of Teichmüller theory that concerns a certain type of canonical deformation associated to an elliptic curve over a number field and a prime number lgeq 5. We begin our survey of interuniversal Teichmüller theory with a review of the technical difficulties that arise in applying scheme-theoretic Hodge-Arakelov theory to diophantine geometry. It is precisely the goal of overcoming these technical difficulties that motivated the author to construct the nonscheme-theoretic deformations that form the content of inter-universal Teichmüller theory. Next, we discuss generalities concerning "Teichmüller-theoretic deformations" of various familiar geometric and arithmetic objects which at first glance appear one-dimensional, but in fact have two underlying dimensions. We then proceed to discuss in some detail the various components of the log-theta-lattice, which forms the central stage for the various constructions of inter-universal Teichmüller theory. Many of these constructions may be understood to a certain extent by considering the analogy of these constructions with such classical results as Jacobi' s identity for the theta function and the integral of the Gaussian distribution over the real line. We then discuss the "inter-universal" aspects of the theory, which lead naturally to the introduction of anabelian techniques. Finally, we summarize the main abstract theoretic and diophantine consequences of inter-universal Teichmüller theory, which include a verication of the ABC/Szpiro Conjecture.