# Part IV - Topics in Number Theory

## Lectured by A. J. Scholl, Lent 2018

These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine.

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# Contents

- V Full version
- 0 Introduction
- 1 Class field theory
- 1.1 Preliminaries
- 1.2 Local class field theory
- 1.3 Global class field theory
- 1.4 Ideal-theoretic description of global class field theory
- 2 Hecke characters and abelian L-functions
- 3 Abelian L-functions
- 4 Non-abelian L-functions
- 5 ℓ-adic representations
- 6 The Langlands correspondence
- 6.1 Representations of groups
- 6.2 Hecke algebras
- 6.3 The Langlands classification
- 6.4 Local Langlands correspondence
- 7 Modular forms and representation theory