Module Details |
The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module. |
Title | Elliptic curves | ||
Code | MATH444 | ||
Coordinator |
Dr V Guletskii Mathematical Sciences vladimir.guletskii@liverpool.ac.uk |
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Year | CATS Level | Semester | CATS Value |
Session 2024-25 | Level 7 FHEQ | Second Semester | 15 |
Aims |
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To provide an introduction to the problems and methods in the theory of elliptic curves. To investigate the geometry of ellptic curves and their arithmetic in the context of finite fields, p-adic fields and rationals. To outline the use of elliptic curves in cryptography. |
Learning Outcomes |
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(LO1) The ability to describe and to work with the group structure on a given elliptic curve. |
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(LO2) Understanding and application of the Abel-Jacobi theorem. |
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(LO3) To estimate the number of points on an elliptic curve over a finite field. |
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(LO4) To use the reduction map to investigate torsion points on a curve over Q. |
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(LO5) To apply descent to obtain so-called Weak Mordell-Weil Theorem. |
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(LO6) Use heights of points on elliptic curves to investigate the group of rational points on an elliptic curve. |
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(LO7) Understanding and application of Mordell-Weil theorem. Encode and decode using public keys. |
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(S1) Problem solving skills |
Syllabus |
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Preliminary review of topics from Algebra, Analysis, Number Theory and Geometry. |
Recommended Texts |
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Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module. |
Pre-requisites before taking this module (other modules and/or general educational/academic requirements): |
MATH102 CALCULUS II; MATH101 Calculus I; MATH103 Introduction to Linear Algebra; MATH101 Calculus I; MATH103 Introduction to Linear Algebra; MATH102 CALCULUS II; MATH247 Commutative Algebra; MATH244 Linear Algebra and Geometry; MATH244 Linear Algebra and Geometry; MATH247 Commutative Algebra |
Co-requisite modules: |
Modules for which this module is a pre-requisite: |
Programme(s) (including Year of Study) to which this module is available on a required basis: |
Programme(s) (including Year of Study) to which this module is available on an optional basis: |
Assessment |
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EXAM | Duration | Timing (Semester) |
% of final mark |
Resit/resubmission opportunity |
Penalty for late submission |
Notes |
written exam | 120 | 50 | ||||
CONTINUOUS | Duration | Timing (Semester) |
% of final mark |
Resit/resubmission opportunity |
Penalty for late submission |
Notes |
Homework 5 | 0 | 10 | ||||
Homework 4 | 0 | 10 | ||||
Homework 3 | 0 | 10 | ||||
Homework 2 | 0 | 10 | ||||
Homework 1 | 0 | 10 |