MAGIC049: Modular Forms

Course details

A specialist MAGIC course

Semester

Autumn 2022
Monday, October 3rd to Friday, December 9th

Hours

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

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

No bibliography has been specified for this course.

Assessment

The assessment for this course will be released on Monday 9th January 2023 and is due in by Saturday 21st January 2023 at 23:59.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.

Files

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Lectures

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