A Panoramic Overview of Inter-universal Teichmüller Theory

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 l ≥ 5. We begin our survey of inter-universal 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 non-scheme-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 construc-tions 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 distribu