IUT as anabelian gateway to diophantine geometry and analytic number theory - S.Mochizuki
Abstract
In this talk, we address this question from various points of view. First, we discuss the fundamental framework underlying the relationship established by IUT between anabelian geometry, on the one hand, and diophantine geometry/analytic number theory, on the other. This discussion centers around the N-th power map on a subring of a field and the difference between regarding a group as a Galois group and as an abstract group that is not equipped with an embedding into the automorphism group of a field. Here, we emphasize that this discussion is entirely elementary and only assumes a knowledge of groups/monoids, rings, fields, and the elementary geometry surrounding the projective line/Riemann sphere. We then proceed to survey recent developments (work in progress) in IUT, namely, the Galois-orbit version of IUT, which has new applications to the Section Conjecture (in anabelian geometry) and the nonexistence of Siegel zeroes of certain Dirichlet \(L\)-functions. The application to the Section Conjecture is interesting in that it exhibits and reconfirms the essentially anabelian content of IUT, i.e., as a theory based on anabelian geometry that is applied to prove new results in anabelian geometry. On the other hand, these recent applications, taken together with the original application of IUT to the ABC/Szpiro/Vojta Conjectures, are also noteworthy in that they may be regarded as a striking example of Poincaré's famous quote to the effect that
``mathematics is the art of giving the same name to different things''.
That is to say, the common name ``IUT'' that may be regarded as describing, in essence, a single mathematical phenomenon that manifests itself, depending on relatively inessential (!) differences of context, as various (at first glance, unrelated!) diverse phenomena in anabelian geometry, diophantine geometry, and analytic number theory. The relationship with Poincaré's famous quote is also fascinating in that it was apparently motivated by various mathematical observations on the part of Poincaré concerning the similarities between transformation group symmetries of modular functions such as theta functions and symmetry groups of the hyperbolic geometry of the upper half-plane --- all of which are topics (cf. the discussion above of Galois groups versus abstract groups!) that bear a profound relationship to IUT.