Universität Wien

250113 VO Topics in Number Theory (2023S)

3.00 ECTS (2.00 SWS), SPL 25 - Mathematik

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Details

Language: English

Examination dates

Lecturers

Classes (iCal) - next class is marked with N

  • Wednesday 01.03. 14:15 - 15:45 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 08.03. 14:15 - 15:45 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 15.03. 14:15 - 15:45 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 22.03. 14:15 - 15:45 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 29.03. 14:15 - 15:45 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 19.04. 14:15 - 15:45 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 26.04. 14:15 - 15:45 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 03.05. 14:15 - 15:45 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 10.05. 14:15 - 15:45 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 17.05. 14:15 - 15:45 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 24.05. 14:15 - 15:45 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 31.05. 14:15 - 15:45 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 07.06. 14:15 - 15:45 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 14.06. 14:15 - 15:45 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 21.06. 14:15 - 15:45 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 28.06. 14:15 - 15:45 Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock

Information

Aims, contents and method of the course

In this course we will study aspects of the representation-theory of the group GL(n) over the field of rational numbers Q and its interconnections to modern number theory, notably by investigating a broad and conceptual generalization of the Riemann zeta function, modular forms and class field theory.

We will start off be introducing the (topological) ring of adèles over Q. Then we will pass to its group of units, called idèles, and study its irreducible representations. Attached to those are L-functions, called "Hecke L-functions", which encode by analytic means large parts of the arithmetic of Q -- the "easiest" such Hecke L-function being the Riemann zeta function.

Then we will consider the matrix-group GL(n) over Q, which itself provides a generalization of the previous setup (the case of idèles being the case of GL(1)). The relevant representations of GL(n) are in general of infinite dimension and called "automorphic". Their attached L-functions, now to be called "automorphic L-functions", are mysterious objects of central importance in today’s research: Indeed, the case of GL(2) just recovers the full theory of modular forms and their attached L-functions. We will put some spotlights on the internal properties of automorphic representations of GL(n) and their attached L-functions and how they relate to arithmetic.

Prerequisites: basic concepts from (algebraic) number theory, from topology, and basic knowledge from real analysis in several dimensions.

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

J. Neukirch "Algebraic Number Theory" Springer (1999)
D. Goldfeld, J. Hundley, "Automorphic Representations and L-Functions for the General Linear Group" I & II, Cambridge Univ. Press (2011)
D. Bumb, "Automorphic Forms and Representations" Cambridge Univ. Press (1998)

Association in the course directory

MALV

Last modified: Tu 12.09.2023 08:07