### Introduction to Inter-universal Teichmuller theory I

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 so-called theta-pilot object, whose very construction relies on the symmetries aforementioned.

It is fair to say that the main construction of IUT is the so-called multiradial (i.e. invariant under changes of ring structures) representation of the theta-pilot object. The construction of such a representation can only be achieved under the indeterminacies/equivalences (Ind1, 2, 3), which concern the automorphisms of local Galois groups, automorphisms of local unit groups, and change of additive structures of local integer rings, respectively. The multiradial representation is in particular compatible with the crucial theta-links between different copies of Hodge theaters, which dismantle ring structures so as to extract information on arithmetic degrees. The use of theta-link was motivated by the previous works of Mochizuki, such as Hodge-Arakelov theory. It follows essentially from this compatibility that the arithmetic line bundle associated to the multiradial representation has degree close to zero, as was predicted by Hodge-Arakelov theory. This is the main result of IUT theory. Such a result and some standard techniques finally lead to the proof of ABC conjecture.