题目：Introduction to Inter-universal Teichmuller theory  

报告人：谭福成 Fucheng Tan（京都大学数学研究所Research Institute for Mathematical Sciences, Kyoto University）

摘要：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 link to  theta functions via anabelian geometry, 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 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 of geometric nature since the corresponding set of cusps is naturally a subquotient of the geometric fundamental group of a hyperbolic curve determined by Eand l. The first symmetry will be applied to copies ofthe field of moduli of E, while the second 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 everything 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 ring 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. This is the main result of IUT theory. Such a result and some standard techniques finally lead to the proof of ABC conjecture.

时间：  4月4日（周四）下午15:00-17:00

4月8日（周日）下午14:00-16:00

4月9日（周一）下午14:00-16:00

地点：首师大新教二楼827教室

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