## Topics in Arakelov Geometry

**Speaker(s): ** C.Soulé (CNRS and IHES)

**Time: ** 00:00-00:00 July 20, 2011 - July 29, 2011

**Venue: ** Room 1328 at BICMR, Resource Plaza, Peking University

Title： Topics in Arakelov Geometry

Speaker：C.Soulé (CNRS and IHES)

Time: 7.20,7.22,7.27,7.29 14:00--16：00

Venue: Room 1328 at BICMR, Resource Plaza, Peking University

Abstract: Arakelov theory studies arithmetic varieties, i.e. algebraic varieties defined on the integers, in a way analogous to classical algebraic geometry of varieties over a field. It is a powerful tool to bound the size of solutions of diophantine equations.

In the first two talks, we shall consider an arithmetic variety X equipped with an hermitian line bundle. To any closed subset Y in X we attach a real number, called the height of Y. It can be defined by induction on the dimension of Y.

In the last two talks, which are quite independent of the previous ones, we suppose that X has dimension two.

Following Parshin and Moret-Bailly, we state a conjecture on the height of X (with respect to a suitable hermitian line bundle) which would imply the abc conjecture.

We then show that this height is bounded from above by the successive minima of a euclidean lattice.

The first two talks require a basic knowledge of scheme theory. The last ones are more of a survey, and they will contain fewer proofs.