The aim of this course is to familiarize students with basic concepts, techniques, and applications of modular form theory as well as with some modern (deep) results about modular forms and their applications. The students will also learn how to perform explicit calculations with modular forms using the (free open-source) mathematics software system Sage.

**Description**

This is an introductory course into the subject of modular forms and their applications. A modular form is a complex analytic function defined on the complex upper half plane which has a certain symmetry with respect to the action of SL(2,Z) (or some subgroup) on the upper half plane and which satisfies some growth condition. Near the turn of the 19th to 20th century it became clear that the coefficients of series expansions of modular forms are often integers of significant number-theoretical interest. This insight has only grown in the 20th century; nowadays modular forms are an indispensable tool in modern number theory (but also play an important role in other subjects, e.g. physics). One of the great successes was their use in the proof of Fermat’s Last Theorem by Andrew Wiles around 1994.

Topics include: the modular group and congruence subgroups, definition of modular forms and first properties, Eisenstein series, theta series, valence formulas, Hecke operators, Atkin-Lehner-Li theory, L-functions, modular curves, modularity. In the final lecture(s) we will give a global outlook on the use of modular forms (and their associated Galois

representations) in the solution of Diophantine equations, in particular Fermat’s Last Theorem.

**Prerequisites**

Basic knowledge of group theory and complex analysis.

**Hours**

Generally, there will be 2 × 45 minutes of lectures, followed by a one-hour exercise session.

**Examination**

The final mark will be based on regular hand-in exercises and a final exam (written or oral, depending on the number of participants)

**Literature**

Lecture notes will be made available before the start of the course.

The free open-source mathematics software system Sage will be used to perform explicit modular forms computations.

For background reading and more we can also recommend:

F. Diamond and J. Shurman, “A First Course in Modular Forms”, Graduate Texts in Mathematics 228, Springer-Verlag, 2005. (Covers most of what we will do and much more.)

J.-P. Serre, “A Course in Arithmetic”, Graduate Texts in Mathematics 7, Springer-Verlag, 1973. (Chapter VII is very good introductory reading.)

W.A. Stein, “Modular Forms, a Computational Approach”, Graduate Studies in Mathematics, American Mathematical Society, 2007. (Emphasis on computations using the free open-source mathematics software system Sage.)

J.S. Milne, “Modular Functions and Modular Forms”, online course notes.

J.H. Bruinier, G. van der Geer, G. Harder and D. Zagier, “The 1-2-3 of Modular Forms”, Universitext, Springer-Verlag, 2008.

**Instructors**

Bruin, P. (Universiteit Leiden)

Dahmen, S. (Vrije Universiteit)

E-mail p.j.bruin@math.leidenuniv.nl; s.r.dahmen@vu.nl

**Remarks**

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