# MAGIC049: Modular Forms

## Course details

A specialist MAGIC course

### Semester

Spring 2024
Monday, January 29th to Friday, March 22nd; Monday, April 22nd to Friday, May 3rd

### Hours

Live lecture hours
10
Recorded lecture hours
0
40

### Timetable

Mondays
12:05 - 12:55 (UK)

### Course forum

Visit the MAGIC049 forum

## 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

• JS

### Jack Shotton

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.

• A Course in Arithmetic (J-P. Serre, book)
• A First Course in Modular Forms (Fred Diamond and Jerry Shurman, book)

## Assessment

The assessment for this course will be released on Monday 13th May 2024 at 00:00 and is due in before Friday 24th May 2024 at 11:00.

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 hoursâ€™ work to attain the pass mark, which is 50%.

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