MAGIC049: Modular Forms

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.

Good command of complex analysis and algebra.

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

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
Theta series, arithmetic applications
Modular forms and Dirichlet series, functional equation
Hecke operators, Petersson scalar product

MAGIC049: Modular Forms

Course details

Semester

Hours

Timetable

Description

Prerequisites

Syllabus

Lecturer

Jack Shotton

Bibliography

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Assessment

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Lectures