MAGIC049: Modular Forms

Course details

A specialist MAGIC course

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 (UK)

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

    Professor Jens Funke

    University
    Durham University

Bibliography

Follow the link for a book to take you to the relevant Google Book Search page

You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library' - this sometimes works well, but not always - you will need to enter your location, but it will be saved after you do that for the first time.

Assessment

Attention needed

Assessment information will be available nearer the time.

Files

Only current consortium members and subscribers have access to these files.

Please log in to view course materials.

Lectures

Please log in to view lecture recordings.