/sci/ - Science & Math

Three upcoming lectures:
http://math.sustc.edu.cn/event/10808.html 2018-01-26
http://math.sustc.edu.cn/event/10809.html 2018-01-26
http://math.sustc.edu.cn/event/10810.html 2018-01-29

Introduction to Inter-universal Teichmuller theory I/II/III

In this series of talks, we will explain the main result and some crucial technical points of the Inter-universal Teichmuller (aka IUT) theory of Shinichi Mochizuki. In the end, we also give a sketched proof of the ABC/Vojta conjecture (for hyperbolic curves), as an application of IUT theory.

In IUT, one starts with a suitable elliptic curve E over a number field F and a prime number l (among other technical data), and studies such a collection of data via certain hyperbolic curves, which are used in the theory of etale theta function. In particular, anabelian geometry (for hyperbolic curves) and etale theta function form the foundation of IUT.

A variety of geometric and arithmetic information about the elliptic curve and theta function is recorded in the so-called Hodge theater. More concretely, a Hodge theater is designed to carry two kinds of symmetries associated to a fixed quotient of l-torsions of the elliptic curve, which are represented by the cusps of certain hyperbolic curves. One of them is called the multiplicative symmetry, which is of arithmetic nature as the corresponding set of cusps is naturally a subquotient of the absolute Galois group of the field of moduli of E. The other is called the additive symmetry, which is of geometric nature since the corresponding set of cusps is naturally a subquotient of the geometric fundamental group of a hyperbolic curve determined by E and l. The multiplicative symmetry will be applied to copies of (Frobenioids associated to) the the field of moduli of E, while the additive symmetry assures that the conjugacies of local Galois groups on various values of theta function (at these cusps) are synchronized. These theta values and the number field will determine the...