Department of Mathematics

Joachim Wehler

LMU MÃ¼nchen

Winter term 2020/21

Modular forms and elliptic curves (4+2)

Winter term 2020/21

## Description

## Lecture notes

## References

## Problem session

## Some PARI scripts

## Examination

## 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 S^{1}xS^{1}. The moduli space of complex tori
is the orbit space of the operation of the modular group Î“=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 Î“, modular curves, Ramanujanâ€™s Ï„-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 notesThe 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 â€žPart II. Analytic Methodsâ€œ.

**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, AlexeÃ¯:**
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.

## 5. Some PARI scripts

**Lecture notes:**

**Problem sheets:**

## 6. Examination

The oral examination ("ModulprÃ¼fung") via Zoom is scheduled for 23.2.2021 and 24.2.2021.

The examination will cover the topics from the lecture and from the problem sheets.

It will take 30-60 minutes for each participant.

The use of written notes or electronic files or devices is not admitted during the examination.

Because the examination will be held via Zoom each participant has to fill in the file

Each participant is asked to send me a signed copy until**Monday, 15.2.2021**via email.

I will send you a Zoom-Link according to the following schedule some minutes before the examination starts:

**23.2.2021**

9.00-10.00 Mr. L. Gambarte

10.15-11.15 Mrs. E. Hsiao

11.30-12.30 Mr. R. Mader

**24.2.2021**

9.00-10.00 Mr. K. Mattis

10.15-11.15 Mrs. D. Nkeng

11.30-12.30 Mr. K. Ã–ztas