In this course, we study the rational points on curves over finite fields : they are solutions of some polynomial equations and we are particularly interested in their number. We introduce and use various tools to give bounds on the number of such points : algebraic geometry, zeta functions, …
We motivate this study by explaining various applications (to coding theory, to cryptography, to exponential sums, …) and we give many examples. We will also investigate some statistical aspects, such as the average number of points on certain families of curves.
The homework assignments will count for 25% of the final grade
The midterm exam will contribute a further 25%
And the final exam will account for the remaining 50%
Algebra 1, 2, 3.
Handouts of slides, partial lecture notes and research papers will be provided during the course.