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Elliptic Curves (DIAMANT)
| Credits | 8 credit points |
| Instructors | Cornelissen, G.L.M. (Universiteit Utrecht), Stevenhagen, P. (Universiteit Leiden) |
| g.l.m.cornelissen@math.uu.nl, psh@math.leidenuniv.nl | |
| Aim | 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. |
| Description | 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 (pari, MWrank); Elliptic curves over finite fields with applications (factoring integers, elliptic discrete logarithms and cryptography); complex multiplication; a survey of modularity of elliptic curves. |
| Organization | Each class there will be three 45 minutes time slots: two lectures and one exercise class. |
| Examination | the final grade will be based exclusively on weekly homework. |
| Prerequisites | linear algebra, groups, rings, fields, complex variables. Literature: 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 made available during the course. |
| Remarks | http://www.math.leidenuniv.nl/~streng/elliptic/index.html |