Curves over Finite Fields

Advanced Bachelor course – Fall 2017 – Universiteit Leiden
Lecturer: Richard Griffon


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, ...

This study will be motivated by explaining various applications (to coding theory, to cryptography, to exponential sums, ...) and we give many examples. Time permitting, we will also investigate some statistical aspects, such as the average number of points on certain families of curves.

Web Site
Lectures Mondays 11:00am–13:00am
Location Room 408 – Snellius Gebouw (Universiteit Leiden)
Tentative Syllabus
  1. Overview of Algebraic Geometry
  2. Curves and their function fields
  3. Counting rational points on curves over finite fields
  4. Zeta functions of curves over finite fields
  5. The Riemann Hypothesis for curves
  6. Bounds on the number of rational points
  7. Applications to character sums, to codes, to cryptography, etc.
  8. Estimates of character sums
  9. Statistics on the number of rational points
Further topics (as time permits):
  • Abelian varieties and jacobians of curves
  • Algorithms to compute the number of rational points
  • Serre problem for small genus
  • Distribution of the number of points
  • Elliptic curves over function fields
Prerequisites Algebra 1,2,3; some basic notions of number theory.
Lecture Notes Lectures notes will be posted here, hopefully shortly after the corresponding lectures:
  • Three homework assignments during the semester (40% of final grade),
  • Final exam (60% of final grade).

The exam will take place on Friday, 19th January 2018, from 14:00 to 17:00 (+30 minutes for students with disability), in Room 412 of the Snellius. If you have not yet registered for the exam, please do so quickly. For the exam, you're allowed to bring your notes for the course (handwritten notes, Lecture Notes, Homeworks, ...) but you're not allowed to use electronic devices or books.
The retake will be on Wednesday, 21st February 2018, from 14:00 to 17:00 (+30 minutes for students with disability), in Room 412 of the Snellius.
Literature In addition to the lecture notes, the following books can be consulted:
  • Arithmetic of elliptic curves by J. Silverman (first two chapters),
  • Algebraic geometry in Coding Theory and Cryptography by H. Niederreiter and C. Xing (first four chapters),
  • Algebraic Geometry by R. Hartschorne,
  • Algebraic Curves by W. Fulton,
  • ...
Lecture # Date Topic covered in the lecture Notes Homework
1 Sep 4 Practical info. General introduction and motivation.
Definition of algebraic sets.
I.1.1 --
2 Sep 11 Affine algebraic sets and varieties.
Coordinate rings. Function fields. Dimension.
I.1.1 -- I.1.5 --
3 Sep 18 Projective space, projective sets. I.1.6 -- I.2.2 --
4 Sep 25 Projective varieties, first properties.
Link between affine and projective varieties: projective closure of affines and affine parts of projectives.
I.2.3 -- I.2.7 Read lecture notes.
Try a few exercises in there.
Oct 2 No class
5 Oct 9 Smoothness of curves: definition and various criterions, examples.
Consequences of smoothness. Poles and zeros of functions on curves.
II.1.1 -- II.1.3
6 Oct 16 Order of poles and zeroes of rational functions.
Rational points, places, finiteness results.
Definition of the zeta function.
II.1.4 -- III.3.2 First Homework assignment
7 Oct 23 Divisors on curves. Divisors of rational functions.
Picard group. Link with the zeta function.
Definition of Riemann-Roch spaces.
IV.1 -- IV.2
8 Oct 30 Weak Riemann-Roch theorem. Finiteness of the Picard group of degree 0.
Application to the rationality of the zeta function.
IV.2 -- IV.3
9 Nov 6 Riemann-Roch theorem, application to functional equation of the zeta function.
Consequences for the computation of zeta functions.
Riemann Hypothesis for curves: statement and proof.
IV.3 -- V.3 Second Homework assignment
Nov 13 No class
10 Nov 20 Coding theory (by Raymond van Bommel) see Notes
11 Nov 27 Elliptic curves and Cryptography (by Raymond van Bommel) see Notes Third Homework assignment
12 Dec 4 Application to bounds on exponential sums, number of points on hyperelliptic curves.
Distribution of squares in finite fields.
VI --
13 Dec 11 Frobenius angles, equidistribution, gonality of curves.
Distribution of Frobenius angles of curves.
Bounds on the number of rational points.
EXAM Jan 19 14:00 - 17:00 in Room 412 -- --
RETAKE Feb 21 14:00 - 17:00 in Room 412 -- --