Teaching plan for the course unit

 

 

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General information

 

Course unit name: Geometrical Methods in Number Theory

Course unit code: 568190

Academic year: 2021-2022

Coordinator: Xavier Guitart Morales

Department: Department of Mathematics and Computer Science

Credits: 6

Single program: S

 

 

Estimated learning time

Total number of hours 150

 

Face-to-face and/or online activities

60

 

-  Lecture

Face-to-face

 

45

 

(45, presenciales o no presenciales)

 

-  Practical exercises

Face-to-face

 

15

 

(15, presenciales o no presenciales)

Supervised project

55

Independent learning

35

 

 

Recommendations

 


This course is independent from previous courses. To achieve the course objectives it is recommended that students have prior knowledge of elementary number theory and expertise in the basic concepts of Galois theory.


Further recommendations


All bibliographic references are in English, hence a basic knowledge of this language is required.

 

 

Competences to be gained during study

 


— Capacity to understand the basic principles of the arithmetic of elliptic curves and the determination of their modularity. 
 
— Capacity to solve concrete diophantine problems. 

 

 

 

 

 

Learning objectives

 

Referring to knowledge


— To understand and use arithmetic geometry methods to the study of diophantine problems.
— To understand the demonstration of the theorems on rational points of elliptic curves.
— To learn to work with classical modular forms and the Hasse-Weil L-function.
— To study Frey elliptic curves and their relation with Fermat Last Theorem.

 

Referring to abilities, skills


— To be able to start research in the subject.
— To present mathematical topics in public.
— To learn to use basic bibliographical sources for the subject.

 

 

Teaching blocks

 

1. Arithmetic of elliptic curves

2. Weierstrass equations; The group law

3. Rational points in elliptic curves

4. Hasse-Weil L-function

5. Classical modular forms

6. Applications

 

 

Teaching methods and general organization

 

The methodology of the course includes:

 — Attendance to lectures.

— Attendance to the practical sessions.

— Supervised individual work.

— Independent learning.

The teaching is based on two sessions of two hours each per week during one semester. Throughout the course, students will present orally and deliver in written form some questions and exercises which complement the contents of the lectures. Some of these exercises can be done during class hours.

At the end of the course each student will deliver an oral presentation on a topic related to the syllabus of the course, which will be previously assigned to her/him.

In case of special circumstances that require online teaching, part of the teaching will take place through UB’s Campus Virtual.

 

 

 

Official assessment of learning outcomes

 

In order to pass the course unit, students should show that they have assimilated the contents of the course. The resolution of questions and exercises allows lecturers to assess the ability of students to apply the acquired knowledge. The presentation of the chosen topic allows lecturers to assess the competence in independent learning and also mathematical research skills

If required by special circumstances, the exercises will be sent to the students through Campus Virtual and solutions will also have to be uploaded to the Campus Virtual. The presentation of the special topics, if required due to special circumstances, may be done online.

The final grade takes into account the level of accomplishment of the exercises and questions delivered during the course as well as the work on the assigned topic

 

Examination-based assessment

Students not willing to follow the continuous assessment method must inform the course lecturer during the first two weeks of class