MAGIC049: Modular Forms

Course details

Semester

Autumn 2020
Monday, October 5th to Friday, December 11th

Hours

Live lecture hours
10
Recorded lecture hours
0
Total advised study hours
40

Timetable

Wednesdays
12:05 - 12:55

Course forum

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Description

Modular forms (and automorphic forms/representations) play an increasingly central role in modern number theory, but also in other branches of mathematics and even in physics.

This course gives an introduction to the subject. Here is a sample of topics we plan to cover: 
  • Modular curves, also as Riemann surfaces and as moduli space of elliptic curves (over C); 
  • Modular functions and forms, basic properties, Eisenstein series, eta-function; 
  • Hecke operators, Petersson scalar product; 
  • Modular forms and Dirichlet series, functional equation; 
  • Theta series, arithmetic applications; 
 There are now several good introductory texts on modular forms (each with somewhat different focus) such as:
  •  A First Course in Modular Forms by Diamond and Shurman
  • Topics in Classical Automorphic Forms by Iwaniec
  • Introduction to Elliptic Curves and Modular Forms by Koblitz, and
  • Modular Forms by Miyake
Of course there is also the classical text by Serre and the 1971 book by Shimura. 

Prerequisites

Good command of complex analysis and algebra.

Occasionally, some knowledge of algebraic number theory and Riemann surface theory would be helpful. 

Syllabus


  1. Modular curves, also as Riemann surfaces and as moduli space of elliptic curves (over C) 
  2. Modular functions and forms, basic properties, Eisenstein series, eta-function 
  3. Theta series, arithmetic applications 
  4. Modular forms and Dirichlet series, functional equation 
  5. Hecke operators, Petersson scalar product

Lecturer

  • JF

    Prof Jens Funke

    University
    Durham University

Bibliography

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Assessment

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