## Lecture Notes

Here is a pdf-file of lecture note: it describes the algebraic fundamental groups and Galois representations, how to compute it (added on 2005/3/6), some figures (added on 3/8) Lie algebraization of fundamental groups and Soule's cocycle (added on 3/9). lecture note in pdf.

## Basic References

I am going to lecture on the algebraic fundamental groups. The Book for this subject is SGA1, i.e. Seminaire de Geometrie Algebrique 1 "Revetement Etales et Groupe Fondamental" by A. Grothendieck and Mme M. Raynaud, 1971. This book is carefully written and is not very difficult to read. However, students may feel it too abstract.

A concrete introductory article is a paper by Y. Ihara at ICM 90: "Braids, Galois Groups, and Some Arithmetic Functions", Proceedings of ICM 90, PP.99-120. This paper is compactly written and contains still-open problems, but at many places it seems too compact. There are stated many facts with "it is easy to see", but at least for me, most of them are not easy. Motivated students may read this paper, and try the references there.

## Lecture plan

first announced plan (in pdf)

My present plan is to lecture on the basic facts on algebraic fundamental groups, and then on problems arising from its Lie algebraization, such as those explained in Section 5 of Ihara's ICM 90 paper.

I would like to assume that the audiences have some knowledge on usual topology, namely, classical fundamental groups, unramified coverings, and universal coverings. I would like to assume some basic knowledge on Galois theory of field extension. It is desirable to know the definition of projective limits, profinite completion of groups: for example, I recommend to read PP.116-121 of "Algebraic Number Theory," J.W.S. Cassels and A. Frohlich Ed., Academic Press 1967. I shall use the basic terminologies from category theory. Please learn the definition of categories, functors and the natural transformation (=morphisms between functors). The following technical terms are related to my lecture. My assumption is that most of students know only little about the following terminology, but it would be good to list them here: students may learn some of them in advance, for example by looking at dictionaries like Encyclopedic dictionary of mathematics. Vol. I-IV, MIT Press, Cambridge, MA, 1987.

• A. Algebra.
• groups, projective limits, profinite completion of groups
• lower central series of groups
• free groups
• group cohomology
• completed group algebra
• Galois groups, infinite Galois theory
• inertia groups, ramification
• cyclotomic character
• Lie algebras
• category theory
• representable functors
• Yoneda's Lemma
• schemes
• etale covering
• projective line minus three points
• B. Topology
• fundamental groups in a classical sense (i.e. path modulo homotopy)
• unramified covering, universal covering
• monodromy
• analytic continuation of melomorphic functions
Don't be scared by these terminologies: part of the purpose of my lecture is to get acquainted with these objects.

(To say the truth, I myself am very scared to give a lecture course in such an international school, in English!)

Please do not hesitate to take correspondence by email:

m-mat "at-mark" math.sci.hiroshima-u.ac.jp
See you soon. Makoto.