This is a Mastermath course.

Time and Place: Tuesday 14:00 - 16:45, Utrecht University, see the Program below for the rooms.

First lecture: February 9

Teachers: Peter Bruin <p.j.("last name")@math.leidenuniv.nl> and Sander Dahmen <s.r.("last name")@vu.nl>

Assistants: Raymond van Bommel <R.van.("last part of last name")@math.leidenuniv.nl> and Manolis Tzortzakis <e("last name")@gmail.com>

ECTS Credits: 8

Form of tuition: Generally, there will be 2 x 45 minutes of lecturing, followed by a one-hour exercise session.

Examination: The final mark will for 40% be based on regular hand-in exercises (only the ten highest homework marks are taken into account) and for 60% on a final written exam, with the extra rule that in order to pass the course the student needs to score at least a 5.0 on the final exam.

Prerequisites: Basic knowledge of group theory (groups, (normal) subgroups, cosets, quotients, first isomorphism theorem, group actions) and complex analysis (holomorphic/meromorphic functions, Taylor/Laurent series, path integrals, the residue theorem). These concepts are covered by most introductory courses in group theory and complex analysis, respectively, and can be found in almost any standard text on (abstract) algebra and complex analysis, respectively (see e.g. texts by Ahlfors or Lang).

Aim: 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 SageMath.

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.

- Lecture notes (chapters 1 till 6).
- Extra notes on elliptic curves, modularity, and the conjecture of Birch and Swinnerton-Dyer.
- The free open-source mathematics software system SageMath will be used to perform explicit modular forms computations.
- Intro SageMath (or download the worksheet), last updated on
**May 17**.

- 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.) See also William A. Stein's Modular Forms Database as well as The L-functions and Modular Forms Database.
- 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.

Exercises with a star (*) have to be handed in before/at the start of the next lecture. You may hand them in by email to the assistants. In this case, the solutions must be typeset (e.g. using latex) and in pdf format, unless you agreed in advance to some other form with the assistants.

Covered. 1.1, 1.2, 2.1 (mostly), 2.2, 2.3 (only results)

Exercises. from the notes: 1.1*, 1.2, 2.2; Problem Sheet 1: 1, 2*, 3

Covered. 1.3, rest of 2.1, rest of 2.3, 2.4, 2.5 (delta)

Exercises. from the notes: 2.1, 2.3, 2.4*; Problem Sheet 2: 1, 2, 3*

Covered. 2.5 (j-invariant), 2.6, 2.7, 2.8 (read Thm. 2.12 and Cor. 2.13 yourself)

Exercises. Problem Sheet 3: 1, 2*, 3, 4, 5*, 6

Covered. 3.1, 3.2 (up to the definition following Exercise 3.6), 3.4 (up to Theorem 3.6)

Exercises. from the notes: 3.1, 3.2, 3.3*, 3.4, 3.5, 3.6, 3.9; Problem sheet 4: 1, 2*, 3, 4

Covered. rest of 3.2, 3.3, 3.5, 3.6

Exercises. from the notes: 3.7, 3.8*, 3.10, 3.11; Problem sheet 5: 1*, 2

Covered. rest of 3.4, 3.7, 3.8

Exercises. from the notes: 3.12, 3.13; Problem sheet 6: 1, 2*, 3*, 4

Covered. 4.1, 4.2

Exercises. from the notes: 4.3; Problem sheet 7: 1*, 2, 3, 4*

Covered. 4.3, 4.4 (other proof that Tp commutes with diamond operator), 4.5

Exercises. Problem sheet 8: 1*, 2, 3*, 4, 5

Covered. 4.6, intro SageMath (see Course material above)

Exercises. Problem sheet 9: 1, 2*, 4, 5*

Covered. 5.1, 5.2 (up to Proposition 5.9), 5.3 (definition of old/new subspaces)

Exercises. Problem sheet 10: 1, 2*, 3, 4, 5*

Covered. rest of 5.2, 5.3 (except the proof of Theorem 5.13)

Exercises. Problem sheet 11: 1*, 2, 3, 4, 5*

Covered. 5.3 (proof of Theorem 5.13), Chapter 6

Exercises. from the notes: 6.1*; Problem sheet 12: 1, 2, 3*

Covered. Elliptic curves, modularity, and the conjecture of Birch and Swinnerton-Dyer

Exercises. Practice Exam: This may be handed in, and will count as one of the ten highest homework marks, replacing the lowest of these if the mark for the practice exam is higher, if handed in individually (no collaboration allowed) before the start of the next lecture. You can find solutions here.

Covered. Fermat's Last Theorem (this is not exam material, notes containing more background, amongst other things, can be found here)

Exercises. none