MATH40880

Learning Outcomes:

The student is expected to develop an understanding of the following concepts:

(i) Gauss' reciprocity law and classification of conics over finite fields

(ii) Point counting over finite fields (Chevalley-Warning Theorem).

(iii) p-adic numbers, basic p-adic analysis, multiplicative structure of p-adic fields, Hensel's Lemma, conics over p-adic fields and Hilbert symbol.

(iv) Hasse-Minkowski theorem and classification of conics over number fields.

The module will cover one of the following 2 threads:

(v) An introduction to elliptic curves (group structure and Mordell-Weil theorem).

(vi) Ideal class group and the class number formula.

Indicative Module Content:

Gauss' reciprocity law and classification of conics over finite fields.
Point counting over finite fields.
p-adic numbers, basic p-adic analysis, multiplicative structure of p-adic fields, Hensel's Lemma.
Conics over p-adic fields and Hilbert symbol.
Hasse-Minkowski theorem and classification of conics over number fields.
An introduction to elliptic curves (group structure and Mordell-Weil theorem).
Ideal class group and the class number formula.

Student Effort Hours:

Approaches to Teaching and Learning:

Requirements, Exclusions and Recommendations

MATH30400 & MATH30030, or the consent of the module coordinator.

Also advisable: MATH30090 & MATH30040.

Module Requisites and Incompatibles

Assessment Strategy

Feedback Strategy/Strategies

• Feedback individually to students, post-assessment
• Group/class feedback, post-assessment
• Self-assessment activities

How will my Feedback be Delivered?

Self-assessment activities are tentative and will be based on the following model: Students will be asked to grade their responses to midterm exams, based on a detailed solution sheet. They will prepare a detailed report on their weaknesses in key aspects of the module. Based on the accuracy of this report, their homework grades will be positively updated. The other two feedback strategies to employ (Feedback individually to students, post-assessment and Group/class feedback, post-assessment) are self-explanatory.