## Module Handbook

• Dynamischer Default-Fachbereich geändert auf MAT

# Module MAT-40-27-M-6

## Module Identification

Module Number Module Name CP (Effort)
MAT-40-27-M-6 Elliptic Curves in Positive Characteristics 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-40-27-K-6
Elliptic Curves in Positive Characteristics
P 28 h 62 h - - PL1 3.0 irreg.
• About [MAT-40-27-K-6]: Title: "Elliptic Curves in Positive Characteristics"; 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: 86199 ("Elliptic Curves in Positive Characteristics")

## Contents

• plane projective cubics over arbitrary fields, rational points,
• endomorphisms of elliptic curves, isogenies, complex multiplication, moduli,
• counting of rational points, Hasse bound, Weil conjectures, Schoof's algorithm,
• special algorithms, discrete logarithm for elliptic curves, factorisation.

## Competencies / intended learning achievements

Using the example of elliptic curves over arbitrary fields, the students have enlarged their abilities to work on interdisciplinary mathematics (in particular, the triangle between geometry, algebraic number theory, and cryptography). They have studied the theoretical principles of cryptographic algorithms and are able to apply these concepts and techniques successfully. They understand the proofs presented in the lecture and are able to comprehend and explain them.

## Literature

• J .Silverman, The Arithmetic of Elliptic Curves.

## Requirements for attendance of the module (informal)

More advanced knowledge of Algebra (e.g. from the courses [MAT-12-21-K-3] or [MAT-12-22-K-3]) are useful but not necessarily required.

None

## References to Module / Module Number [MAT-40-27-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.)