# MATH 5020 – Galois Representations

### Course Description:

Title: Galois Representations

Description:

When we study algebraic equations, Galois theory is a fundamental tool. To each polynomial we can associate a group, its Galois group, and from this group we can deduce a lot of information about its algebraic solutions and, viceversa, we can deduce properties of its Galois group from the roots of the polynomial. The so-called absolute Galois group of a field F is an object that is formed from all Galois groups of polynomials over F, and it is the central object of study of algebraic number theory. In this course, we will explore the absolute Galois of a field through its representations, which are called Galois representations.

Prerequisites: two semesters of abstract algebra and a familiarity with algebraic number theory

Meets: at MONT 419, on Tuesdays and Thursdays, from 11:00 – 12:15am. FIRST TWO WEEKS will be online, via WebEx.

### About the Instructor: Álvaro Lozano-Robledo

I’m a Professor of Mathematics, and the Director of Undergraduate Studies at the University of Connecticut

Office hours: Tuesdays 11-12, Wednesdays 10-11, or by appointment, at MONT 233.

### Enrollment:

The enrollment for this class is by permission for undergraduate students. Send me a message if you would like to enroll.

### Outline, books, and other resources:

The course will cover the following topics:

1. Classical Galois Theory. We will review classical Galois theory using the book “Abstract Algebra“, by David S. Dummit, Richard M. Foote, 3rd edition (or 2nd edition, really). Other references for this section (and some below) include “Galois Theory“, by David Cox, and “Fields and Galois Theory“, by Milne (free!).
1. Field extensions and automorphism groups, in characteristic 0 and characteristic p.
2. The Fundamental Theorem of Galois Theory.
3. Extensions of Finite fields.
4. Composite extensions, simple extensions, the primitive element theorem.
5. Cyclotomic extensions, and the Kronecker-Weber theorem.
6. Galois groups of quadratic and cubic polynomials.
2. Infinite Extensions.
1. Algebraic closures. See this handout by Keith Conrad, and Ch. 6 of Milne’s “Fields and Galois Theory”.
2. Cyclotomic extensions. See Washington’s “Introduction to Cyclotomic Fields”.
3. Maximal 2-elementary extension.
3. Profinite Groups.
1. Inverse/projective limits. See Washington’s Appendix to “Introduction to Cyclotomic Fields”, and Lenstra’s notes.
4. Topological Spaces and Topological Groups. We will draw from Munkres’ “Topology”, Hewitt/Ross “Abstract Harmonic Analysis”, Higgins’ “Introduction to Topological Groups“, and Pontryagin’s “Topological Groups”. See also these notes by Ryan Vinroot.
1. Basic properties of topological groups.
2. The Krull topology.
5. The Fundamental Theorem of Infinite Galois Theory. Milne’s “Fields and Galois Theory”, Ch. 7.
1. Proof.
2. Examples.
6. Review of basic algebraic number theory
1. Splitting of primes in extensions
2. Decomposition and inertia subgroups
3. Frobenius elements
4. Chebotarev density theorem
7. The absolute Galois group
8. Galois representations, basic theory
1. Artin, mod-l, l-adic Galois representations
1. Definition.
2. Examples.
10. Galois representations attached to elliptic curves
1. Classification of l-adic Galois representations attached to elliptic curves
11. Galois representations attached to modular forms
12. Galois representations attached to cohomology groups
13. Deformations of Galois representations

### Suggested Exercises:

Homework will be assigned and collected every other week. Homework will be graded on completion only. If you have questions about how to solve a problem, you should talk to me in office hours or ask questions in class.

The homework problem sets will be posted here (file will be updated frequently):

The grade will be computed based on homework performance and class participation.

The total grade will be computed as follows:

### Course Outline:

DF = Dummit and Foote
G = Gouvea’s “p-adic Numbers: an introduction”
H = Higgins’ “An introduction to topological groups”
KC = Conrad’s handout on algebraic closures.
L = Lenstra’s notes on Profinite Groups.
M = Milne’s “Field and Galois theory”
S = Serre’s “Abelian l-adic Galois representations”
W = Washington’s “Introduction to cyclotomic fields”

 Week Topics Notes Reference 1 Introduction to Galois representations Slides [DF] Ch 14 Intro (part 2), and review of Galois theory (part 1) Slides for intro and Gal. Thy. [DF] Ch 14 2 Review of Galois theory (part 2) Slides [DF] Ch 14 Cyclotomic extensions Slides [DF] Ch 14 3 Galois groups of polynomials of degrees 2, 3, and 4; quadratic characters, cyclotomic characters [DF] Ch 14 Algebraic closures, infinite Galois extensions [KC], and [M] Ch. 6 4 p-adic extensions of Q; p-adic completions of Q [G] p-adic numbers, inverse limits [L] 5 Topological groups (definition and examples), topology on profinite groups [H] Morphisms, quotients, products, fundamental system of neighborhoods [H] 6 Density, products, fundamental systems of neighborhoods Slides [H] Separation axioms (part 1), connectedness, compactness (part 2) Slides [H] 7 Profinite groups and their topology [H] The Krull topology on the Galois group [M] 8 Krull and profinite topology coincide, the Fundamental Theorem of Galois Theory [M] Examples of the Galois correspondence 9 More examples of the Galois correspondence (part 1), Galois theory of the algebraic closure of Q_p (part 2) Slides Structure of the absolute Galois group of Q_p, and introduction to Galois representations (video) Slides [S] 10 Topology of Aut(V) and stable lattices [S] Examples of Galois representations: roots of unity, intro to elliptic curves and their Galois representations Slides [S] 11 Galois representations attached to abelian varieties and cohomology groups Frobenius elements 12 Chebotarev and examples Unramified Galois representations, Frobenius elements and Chebotarev 13 mod-ell, ell-adic, and Artin Galois representaitons Properties of Galois representations attached to elliptic curves 14 mod-ell representations attached to elliptic curves Galois representations attached to CM elliptic curves, and ell-adic Galois representations

### Policy Statements:

Please refer to http://provost.uconn.edu/syllabi-references/ for the common policies we follow at UConn.

• Policy Against Discrimination, Harassment and Inappropriate Romantic Relationships — The University is committed to maintaining an environment free of discrimination or discriminatory harassment directed toward any person or group within its community – students, employees, or visitors.  Academic and professional excellence can flourish only when each member of our community is assured an atmosphere of mutual respect.  All members of the University community are responsible for the maintenance of an academic and work environment in which people are free to learn and work without fear of discrimination or discriminatory harassment.  In addition, inappropriate Romantic relationships can undermine the University’s mission when those in positions of authority abuse or appear to abuse their authority.  To that end, and in accordance with federal and state law, the University prohibits discrimination and discriminatory harassment, as well as inappropriate Romantic relationships, and such behavior will be met with appropriate disciplinary action, up to and including dismissal from the University. (More information is available at http://policy.uconn.edu/?p=2884.)
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• Attendance — Your instructor expects you to attend class regularly. Besides being nearly essential for developing your understanding of the material, your regular attendance in class is good for the morale of the class and is indicative of your interest in the subject and your engagement in the course. You are responsible for the material discussed in class and in the assigned reading in the text.
• Student Conduct Code — Students are expected to conduct themselves in accordance with UConn’s Student Conduct Code.
• Academic Integrity Statement — This course expects all students to act in accordance with the Guidelines for Academic Integrity at the University of Connecticut. Because questions of intellectual property are important to the field of this course, we will discuss academic honesty as a topic and not just a policy. If you have questions about academic integrity or intellectual property, you should consult with your instructor. Additionally, consult UConn’s guidelines for academic integrity.
• Students with Disabilities — The Center for Students with Disabilities (CSD) at UConn provides accommodations and services for qualified students with disabilities. If you have a documented disability for which you wish to request academic accommodations and have not contacted the CSD, please do so as soon as possible. The CSD is located in Wilbur Cross, Room 204 and can be reached at (860) 486-2020 or at csd@uconn.edu. Detailed information regarding the accommodations process is also available on their website at www.csd.uconn.edu.
• Final Exam Policy — In accordance with UConn policy, students are required to be available for their final exam and/or complete any assessment during the time stated. If you have a conflict with this time you must obtain official permission to schedule a make-up exam with the Office of Student Support and Advocacy (OSSA). If permission is granted, OSSA will notify the instructor. Please note that vacations, previously purchased tickets or reservations, graduations, social events, misreading the assessment schedule, and oversleeping are not viable reasons for rescheduling a final.