# Topics In Number Theory: Galois Representations

## Introduction

In number theory the law of quadratic reciprocity gives conditions for the solvability of quadratic equations modulo prime numbers. A large part of number theory, known as the Langlands program, can be thought of as the search for a generalization of quadratic recprocity as it provides a, conjectural but "ultimate", reciprocity law. In modern language this search for reciprocity laws can be rephrased as the study of a single group, the absolute Galois group $${\mathrm {Gal}}(\overline {\mathbb Q}/{\mathbb Q})$$ and its representations.

The goal of this course is to make a first step into this field of mathematics. We will continue right where the course on algebraic number theory has stopped and start with a study of the field $${\mathbb Q}_p$$ its finite extensions and continue with the following topics

• The absolute Galois groups of $${\mathbb Q}$$ and $${\mathbb Q}_p$$. Frobenius elements, ramification.

• Local and global class field theory. Chebotarev density theorem.

• Galois representations, Artin representations.

• Elliptic curves and their associated Galois representations.

Additionally at the end of the course, Tim Dokchitser will give a number of lectures on computational aspects.

Note that the above list is tentative, it may change depending on time and also preferences of the students.

## Lecture format

The lectures will be offered online, live via zoom and ipad. The ipad notes of the lecture are posted here.

## Examination

(this section has not yet been finalized, details may change)

At the end of the course there will be a written, closed book exam with zoom surveillance. Additionally there will be an assignment from the computer lab which will count for $$20\%$$ towards your final.

In the dropbox here I have uploaded a file called Exercises.pdf. Some problems in the exam will consist of a random choice from these exercises. As the semester goes on I will update this file every once in a while. Please let me know if you find any typos or mistakes in these exercises!

## Schedule

The lectures will be on Fridays, from 9:00 AM to 11:00 AM. The first lecture is on the 5th of February.

Due to holidays, there will be no lectures on April 2nd, April 9th, May 7th and May 14th. The exam will be on May 28th.

See here for information about the computer lab sessions (provided by Tim Dokchitser).

If you plan to follow this course, please let me know, so that I can share the zoom link with you.