MATH40530 Special Topics I

Academic Year 2021/2022

Theory of Formal Groups

The module will cover the foundations of the theory of formal groups and its applications (e.g. Local Class Field Theory and the local theory of elliptic curves).

PREREQUISITES: The students should be comfortable with Group and Ring Theory. It is strongly advised that the students have completed an equivalent of MATH30130, as well as Galois Theory.

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Curricular information is subject to change

Learning Outcomes:

The students will be familiar with the general theory of formal groups, with a particular emphasis on Lubin-Tate formal groups and applications to arithmetic geometry.

Indicative Module Content:

Definition and elementary properties of formal groups
The Functional Equation -- Integrality lemma
Universal 1-dimensional commutative formal group law
Logarithms
Invariant differential forms
Most one-dimensional formal group laws are commutative (without proof)
Honda formal group laws
Lubin--Tate formal group laws
Homomorphisms, Endomorphisms, and the classification of formal groups via power series methods
Classification of one-dimensional formal group laws over finite fields (without proofs)
Local Class Field Theory
Zeta functions of elliptic curves over Q and Atkin--Swinnerton-Dyer conjectures
Tate modules

Student Effort Hours: 
Student Effort Type Hours
Lectures

0
Specified Learning Activities

88
Autonomous Student Learning

100
Total

188
Approaches to Teaching and Learning:
Active/Task-based Learning
Lectures
Enquiry & Problem-based Learning 
Requirements, Exclusions and Recommendations

Not applicable to this module.


Module Requisites and Incompatibles
Not applicable to this module.
 
Assessment Strategy  
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Essay: Final project Throughout the Trimester n/a Standard conversion grade scale 40% No

100

Carry forward of passed components
No
 
Resit In Terminal Exam
Autumn No
Please see Student Jargon Buster for more information about remediation types and timing. 
Feedback Strategy/Strategies

• Feedback individually to students, post-assessment
• Peer review activities

How will my Feedback be Delivered?

Individual feedback will be given on the first draft of the project (due Week 8) as well as the final version of the project.

"Formal groups and applications", by Michiel Hazewinkel