Universität Wien
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250006 VO Galois Representations in Number Theory (2024W)

3.00 ECTS (2.00 SWS), SPL 25 - Mathematik
We 22.01. 13:15-14:45 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock

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Details

Language: English

Lecturers

Classes (iCal) - next class is marked with N

  • Wednesday 02.10. 13:15 - 14:45 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 09.10. 13:15 - 14:45 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 16.10. 13:15 - 14:45 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 23.10. 13:15 - 14:45 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 30.10. 13:15 - 14:45 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 06.11. 13:15 - 14:45 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 13.11. 13:15 - 14:45 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 20.11. 13:15 - 14:45 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 27.11. 13:15 - 14:45 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 04.12. 13:15 - 14:45 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 11.12. 13:15 - 14:45 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 08.01. 13:15 - 14:45 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 15.01. 13:15 - 14:45 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 29.01. 13:15 - 14:45 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock

Information

Aims, contents and method of the course

This lecture is an introduction to the representation theory of absolute Galois groups of algebraic number fields and their eminent role in modern number theory.

We start with basic concepts from higher Galois theory of (countably) infinite-dimensional field extensions (“algebraic Galois extensions”, “absolute Galois group”, “Krull topology”, etc.). In the next step we study fundamental concepts of the representation theory of Galois groups in the general context of Artin representations, l-adic representations and “mod-l” representations.

In the second part of the lecture we want to examine the role of the representation theory of absolute Galois groups in the Langlands program. The cases n=1 and n=2 will serve as a role model here, i.e. the cases of the groups GL(1) and GL(2), in which many prominent, deep results are known, but also the case of general n ≥1 will be discussed in quite detail. This is the most speculative part of the lecture, as many (if not most) of the questions we consider here are still unresolved problems in current research.

Explicitly assumed prior knowledge: Attendance of a lecture on (i) algebraic number theory, (ii) topology and (iii) algebra (especially basic knowledge of finite Galois extensions). Attending my "VO Advanced topics in global number theory" from SS2024 and a lecture on algebraic topology is particularly helpful for a deeper understanding of the topics of the second part of this lecture, but not mandatory.

Assessment and permitted materials

Oral exam at the end of the semester on appointment.

Minimum requirements and assessment criteria

Good knowledge of the central concepts presented in the course, as well as the ability to apply them in certain examples. We apply the usual standards for exams of Master's courses.

Examination topics

The contents presented in the course. Exceptions (if any) would be announced as the course progresses.

Reading list

W. Hungerford, „Algebra“, Springer (1989)
J. Neukirch, „Algebraic Number Theory“, Springer (1999)
A. Schmidt, K. Wingberg, J. Neukirch, „Cohomology of number fields“, Springer, Grundlehren der Mathematischen Wissenschaften (2000)
J.-P. Serre, „Galois Cohomology“, Springer, (1997)

Association in the course directory

MALV

Last modified: We 31.07.2024 11:06