Elliptic Curves

Elliptic curves are fundamental objects in a large part of

mathematics. Along various historical paths, their origins can be

traced to calculus, complex analysis and algebraic geometry, and their

arithmetic aspects have made them key objects in modern cryptography

and in Wiles' proof of Fermat's last theorem. This course is an

introduction to the algebraic, geometric, complex analytic, and number

theoretical aspects of the theory of elliptic curves.

The topics treated include a general discussion of elliptic integrals

and functions, elliptic curves and their group law; Diophantine

equations in two variables; Mordell's theorem, with computational

aspects; Computer class (sage); Elliptic curves over finite fields

with applications (factoring integers, elliptic discrete logarithms

and cryptography); complex multiplication; a survey of modularity of

elliptic curves.

Each class there will be three 45 minutes time slots:

two lectures and one exercise class.

The final grade will be based exclusively on weekly homework.

Newcomers to the subject are suggested to buy the book J.H. Silverman

and J. Tate: Rational Points on Elliptic Curves, Undergraduate Texts

in Mathematics, Springer Verlag, Corr. 2nd printing, 1994, Hardcover,

ISBN: 978-0-387-97825-3: it contains a lot of the material treated in

the course. Advanced students with a good knowledge of algebraic

geometry are recommended to (also) buy J.H. Silverman: The arithmetic

of elliptic curves. Corrected reprint of the 1986 original. Graduate

Texts in Mathematics, 106. Springer-Verlag, New York, 1992. ISBN:

0-387-96203-4.  A complementary literature list and course notes will

be added to the course website during the course.

Linear algebra, groups, rings, fields, complex variables.