Modular forms and elliptic curves (4+2)

Winter term 2020/21

1. Description

Complex analysis, algebra and arithmetic

A topic from the field of complex analysis is the study of 1-dimensional complex tori. They provide different complex structures on the fixed underying topological manifold S¹xS¹. The moduli space of complex tori is the orbit space of the operation of the modular group Γ=SL(2,Z) on the upper half-plane. The compactification of the orbit space is the 1-dimensional complex projective space.

From the viewpoint of algebraic geometry the analogue of tori are elliptic curves. These curves in the projective plane are the zeros of a cubic polynomial (Weierstrass polynomial). Its coefficients determine a number field. This fact allows a refined study of elliptic curves and a posteriori of tori by means of number theory.

Modular forms and the meromorphic modular j-invariant build a bridge between complex analytic geometry and arithmetic geometry. The bridge uncovers remarkable relations between different fields: The uniformization of elliptic curves by modular curves (Proof of the Shimura-Taniyama-Weil conjecture), the relation between the Fourier expansion of j and the irreducible representations of the monster group (Proof of the monstrous moonshine conjecture).

The lecture will introduce to the field and give references to more advanced literature. Some further key words: Weierstrass p-function, Eisenstein series, Hecke congruence subgroups of Γ, modular curves, Ramanujan’s τ-function, 4-squares theorem, Hecke operators.

The course emphasizes the use of the open source software PARI. Hence the lecture also serves as an introduction to computational algebra.

Target audience: Master mathematics or TMP. 9 ECTS (Module WP37, WP36, WP30)

Prerequisites: Complex analysis (Funktionentheorie), Riemann surfaces, sheaf theory. Helpful is some familiarity with algebraic geometry and algebraic number theory.

Scheduled time: Tuesday and Thursday 10-12 am (lecture),Thursday 12-2 pm (problem session).

Update 14.9.2020: The course will be held online via Zoom. Please register by emailing to me. Then I will send you the link to start the Zoom session just before each lecture.

2. Lecture notes

Lecture notes

The lecture notes will be uploaded continuously during the lecture.

3. References (ordered by degree of complexity)

Ahlfors, Lars: Complex Analysis. McGraw-Hill (1966)

Serre, Jean-Pierre: A Course in Arithmetic. Springer (1973). See „Part II. Analytic Methods“.

Tatitscheff, Valdo: A short introduction to Monstrous Moonshine. Preprint (2019)

Milne, James S.: Modular Functions and Modular Forms. (2017)

Zagier, Don: Elliptic Modular Forms and Their Applications.

Diamond, Fred; Shurmann, Jerry: A First Course in Modular Forms. Springer (2005)

Pantchichkine, Alexeï: Formes Modulaire et Courbes Elliptiques.

Zagier, Don: Lecture on Modular Forms (Video)

Software:

PARI: https://pari.math.u-bordeaux.fr/download.html

4. Problem session

In addition to the lecture in class, each week a problem session will be held. The basis is a series of problems for homework. The students are encouraged to discuss their solutions during the problem session.

Problem sheets