Syllabus for Math 8851.

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Syllabus for Math 8851 (Profinite Groups) Fall 2023

### TuTh 11am-12:15 pm, Cabell 038.

- Instructor: Mikhail Ershov
- Office: Kerchof 302
- e-mail: ershov
*at* virginia *dot* edu
- Office hours: TBA
- Course webpage

### Brief course abstract

A profinite group is a topological group isomorphic to the inverse limit of a family of finite groups. (Infinite) profinite groups naturally arise in number theory as these are precisely the Galois groups of infinite Galois extensions. Another common source of examples is algebraic groups over non-Archmimedean local fields whose compact subgroups are profinite (a typical example of such a group
is SL_n(Z_p) where Z_p is the ring of p-adic integers). More generally, profinite groups can be characterized as compact (Hausdorff) totally disconnected groups. Given an arbitrary group G,
one can consider its profinite completion \Ghat, and the canonical map G -> \Ghat is injective whenever G is residually finite. In this way, profinite groups naturally arise in many problems coming from combinatorial and geometric group theory.
The course will roughly consist of three parts:
- A general theory of profinite and pro-p groups with applications to discrete groups.
- Profinite groups in number theory. This part will include a detailed discussion of infinite Galois theory.
- p-adic analytic pro-p groups and the relation with the congruence subgroup property.

If time allows, we will also have a brief discussion of the general theory of locally compact totally disconnected groups.
### References

There is no official text for the course. A substantial portion (but not all) of the material we will discuss is covered by the following books:
*Analytic pro-p groups * by J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal, available electronically through the UVA subscription
*Profinite groups * by J. Wilson
*Profinite groups * by L. Ribes and P. Zalesskii
* Galois theory of p-extensions * by H. Koch
*Lectures on profinite topics in group theory* by B. Klopsch, N. Nikolov and C. Voll, available electronically through the UVA subscription
*New directions in locally compact groups* by P.-E. Caprace and N. Monod, available electronically through the UVA subscription

### Prerequisities

Algebra-I and II and basic general topology. Basic knowledge of algebraic number theory will be helpful, but all the necessary tools will be introduced in the course.

### Homework and Problem Session

- Homework will be assigned regularly (most likely every 10 days or so)

- Approximately every two weeks we will have a problem session
(date and time TBD) where homework problems will be discussed.

- Students taking the class for a grade are expected to
- seriously attempt all the assigned problems
- present homework problems during the problem sessions on the regular basis

### End of the semester Presentations

Each student taking the class for a grade must give a 40-50 min presentation on a chosen topic at the end of the semester.
A list of suggested topics will be provided. You should feel free to choose a different topic (not from the list), but you need to get my approval. In any case, you should inform me about your proposed topic no later than the end of October.
### Add/drop/withdrawal dates:

- Tuesday, September 5 -- Last day to add a course, select the AU (audit) option or change to or from "Credit/No Credit" Option
- Wednesday, September 6 -- Last day to drop a course
- Tuesday, October 17 -- Last day to withdraw from a course

### SDAC

All students with special needs requiring accommodations should present the appropriate paperwork from
the Student Disability Access Center (SDAC). It is the student's responsibility to present this paperwork in a
timely fashion and follow up with the instructor about the accommodations being offered. Accommodations
for test-taking (e.g., extended time) should be arranged at least 5 business days before an exam.