The Fundamental Group of an Algebraic Curve Seminar on Algebraic Geometry, Mit 2002

In this seminar we study geometric properties of algebraic curves, or of Riemann surfaces, with the help of an algebraic object attached: the fundamental group, either the algebraic fundamental group, as introduced by Grothendieck, or the topological fundamental group. Here is the central idea of the seminar. To an algebraic curve X over a eld K we can attach the fundamental group (either in the algebraic context or in the topological vein). This algebraic object gives information about the geometry of X. Commuting between topology and algebraic geometry we can determine rather precisely the structure of this group (at least in characteristic zero, or the prime-top part in characteristic p). This tool enables us to prove theorems, nd structures in arithmetic geometry and in algebraic geometry. Please do not consider these notes as containing complete information on the topics introduced, but rather as a guideline through the literature mentioned. We hope and expect students to work themselves through this beautiful material; and, of course both of us are available whenever you have questions, in case you want to talk about this material, or whatever. In the rst sections we gather together some basic information, connecting topological concepts with deenitions and theorems in algebraic geometry. These are meant as an introduction to, and a basis for studying the following topics, fundamental ideas and theorems: I: Good reduction Given a family (of algebraic curves, of Riemann surfaces, of abelian varieties , or whatever) over a punctured disc, or over the eld of fractions of a discrete valuation ring; try to nd a criterion which ensures that this family can be extended in such a way that the central ber has \good reduction"; it turns out that the Ga-lois representation obtained enables us to formulate precisely this property. In case of abelian varieties this was given by Serre and Tate; in that case the Galois representation, obtained from (co)homology, or from the Tate-`-group, provides the tool necessary. We discuss the case of algebraic curves, where the Galois representation (or the monodromy representation) on the fundamental group of the algebraic curve (sitting over the generic point) provides the necessary information, as Takayuki Oda explained to us, see Section 10.