This course in an introduction to algebraic geometry and arithmetic geometry through the example of elliptic curves, that is non-singular plane projective curves given by a degree 3 equation. A remarkable property of elliptic curves is the existence of a group law on its set of solutions. The first part of the course will be dedicated to the the introduction of the language of algebraic varieties, and more precisely to Hilbert's Nullstellensatz and to projective geometry. Some examples related to Bezout's intersection theorem will be studied. The second part of the course will be dedicated to the properties of plane projective curves and more precisely of elliptic curves. The properties of elliptic curves will be studied over various fields: over the complex numbers, where elliptic curves can be identified to quotients of C by lattices; over finite fields, with Hasse's theorem which provides a bound on the number of points of such elliptic curves; and over the rational numbers, with the celebrated theorem of Mordell. The study of elliptic curves over finite fields will be illustrated with some applications to cryptography and to factoring algorithms. In the case of the field of rational numbers, some explicit examples in which the group of rational points can be computed will be studied.
Bibliography
- 1. J. H. Silverman, Arithmetic of elliptic curves. Graduate Texts in Mathematics, 106. Springer-Verlag, New York, 1986.
- L. C. Washington, Elliptic curves, Number theory and cryptography. Second edition. Discrete Mathematics and its Applications, Chapman & Hall/CRC, Boca Raton, FL, 2008.
- M. Hindry, Arithmétique, Calvage & Mounet, Cambridge University Press, 2008.
- J. H. Silverman et J. Tate, Rational points on elliptic curves, Springer-Verlag, New York, 1992.
Level required: It is advisable to have already manipulated basic algebraic structures (linear algebra, groups, rings, fields, quotients...). In particular, it is recommended to have succcessfully followed the Galois theory course (MAT451).
Language of the course: French or English
- Teaching coordinator: Izquierdo Diego