HU Credits: 4

Degree/Cycle: 2nd degree (Master)

Responsible Department: mathematics

Semester: 2nd Semester

Teaching Languages: Hebrew

Campus: E. Safra

Course/Module Coordinator: Prof Ehud de Shalit

Coordinator Email: rlivne@math.huji.ac.il

Coordinator Office Hours: by appointment

Teaching Staff:
Prof Ehud Deshalit

Course/Module description:
Elliptic functions; the Abel-Jacobi theorem; theta functions; elliptic curves as Riemann surfaces; the embedding into P^2;the Eisenstein series;the addition law; rationality issues; the endomorphism ring;the action of SL(2,Z) on the upper half plane;modular forms;the j-function and the moduli space; cubics and elliptic curves over C; the graded ring of modular forms for SL(2,Z); multiplication by m and the associated exact sequence over C and over Q-bar; Galois cohomology; Hilbert 90; the descent exact sequence; the p-adic analog; reduction mod p; the weak Mordell-Weil theorem; heights; behaviour of heights under a map; the Mordell-Weil theorem;

Course/Module aims:
Acquaintance with the basic properties of elliptic curves over C and modular forms; the basic results over Q and reduction modulo p; the Mordell-Weil theorem.

Learning outcomes - On successful completion of this module, students should be able to:
The students will know the basic theory of elliptic functions, elliptic curves, modular forms and the connection between them; the basic facts about Galois cohomology and its use, together with algebraic geometry, to obtain Diophantine results.

Attendance requirements(%):
100%

Teaching arrangement and method of instruction: frontal lectures. Assignment of exercises

Course/Module Content:
See above

Required Reading:
None

Additional Reading Material:

Course/Module evaluation:
End of year written/oral examination 100 %
Presentation 0 %
Participation in Tutorials 0 %
Project work 0 %
Assignments 0 %
Reports 0 %
Research project 0 %
Quizzes 0 %
Other 0 %

Additional information:
According to the number of participants and their level, there might be a take-home exam or a project work instead of a written examination in class.