## Module Handbook

• Dynamischer Default-Fachbereich geändert auf MAT

# Module MAT-40-19-M-6

## Module Identification

Module Number Module Name CP (Effort)
MAT-40-19-M-6 Elliptic Functions and Elliptic Curves 3.0 CP (90 h)

## Basedata

CP, Effort 3.0 CP = 90 h 1 Sem. irreg. [6] Master (General) [EN] English Decker, Wolfram, Prof. Dr. (PROF | DEPT: MAT) Decker, Wolfram, Prof. Dr. (PROF | DEPT: MAT) Zintl, Jörg, PD Dr. (EXT | DEPT: MAT) [MAT-AGCA] Algebra, Geometry and Computer Algebra [MAT-88.105-SG] M.Sc. Mathematics [NORM] Active

## Courses

Type/SWS Course Number Title Choice in
Module-Part
Presence-Time /
Self-Study
SL SL is
required for exa.
PL CP Sem.
2V MAT-40-19-K-6
Elliptic Functions and Elliptic Curves
P 28 h 62 h - - PL1 3.0 irreg.
• About [MAT-40-19-K-6]: Title: "Elliptic Functions and Elliptic Curves"; Presence-Time: 28 h; Self-Study: 62 h

## Examination achievement PL1

• Form of examination: oral examination (20-30 Min.)
• Examination Frequency: irregular (by arrangement)
• Examination number: 84151 ("Elliptic Functions and Elliptic Curves")

## Contents

• elliptic functions, Weierstrass P-function,
• complex tori,
• plane geometry and geometric construction of a group structure,
• elliptic curves,
• modular forms, modular curves, and classification theory of elliptic curves.

## Competencies / intended learning achievements

Upon successful completion of this module, the students have expanded their knowledge in interdisciplinary mathematics. They have studied elliptic curves, which like a few others, is a class of mathematical objects that is relevant to almost the entire spectrum of mathematics. This ranges from number theory (e.g. Theorem of Fermat) to computational data processing (e.g. encryption algorithms). Exemplarily, the students have learnt how to approach a general problem from a priori very different specialized disciplines (here: using analytic, topological, geometric and/or algebraic methods), and to compare the results across different disciplines. They understand the proofs presented in the lecture and are able to comprehend and explain them.

## Literature

• W. Fischer, I. Lieb: Funktionentheorie,
• G. Fischer: Ebene algebraische Kurven,
• J .Silverman: The Arithmetic of Elliptic Curves.

None

## References to Module / Module Number [MAT-40-19-M-6]

Course of Study Section Choice/Obligation
[MAT-88.105-SG] M.Sc. Mathematics [Core Modules (non specialised)] Pure Mathematics [WP] Compulsory Elective
[MAT-88.706-SG] M.Sc. Mathematics International [Core Modules (non specialised)] Pure Mathematics [WP] Compulsory Elective
Module-Pool Name
[MAT-GM-MPOOL-5] General Mathematics (Introductory Modules M.Sc.)