MAGIC: Modular Forms (MAGIC049)

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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.

Good command of complex analysis and algebra. Occasionally, some knowledge of algebraic number theory and Riemann surface theory would be helpful.

No syllabus information is available yet.

There will be two homework assignments which both need to be passed satisfactorily. HW1: Due March 9 HW2: Due April 13