Description

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

pub.math.leidenuniv.nl/~griffonrmm/CFF.html

Email

r.m.m.griffon@math.leidenuniv.nl

Lectures

Mondays 11:00am–13:00am

Location

Room 408 – Snellius Gebouw (Universiteit Leiden)

Tentative Syllabus

 Overview of Algebraic Geometry
 Curves and their function fields
 Counting rational points on curves over finite fields
 Zeta functions of curves over finite fields
 The Riemann Hypothesis for curves
 Bounds on the number of rational points
 Applications to character sums, to codes, to cryptography, etc.
 Estimates of character sums
 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:
 Detailed bibliography,
 First Chapter from the Lecture notes,
 Second Chapter from the Lecture notes,
 Third Chapter from the Lecture notes,
 The first Homework assignment (which was due on the 6th November).
 Fourth Chapter from the Lecture notes (updated/corrected version, 6th Nov.),
 Fifth Chapter from the Lecture notes,
 The second Homework assignment (which was due on the 20th November).
 Notes for the chapter on Coding Theory (by Raymond van Bommel).
 Notes for the chapter on Elliptic curves and Cryptography (by Raymond van Bommel).
 The third Homework assignment
(which was due on the 11th December).
 Sixth Chapter from the Lecture notes,
 Seventh and Eighth Chapters from the Lecture notes (Chapter 8 is extra, it won't be examined),
 Last year's exam,
 The text of the exam.
 The retake.

Examination

 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,
 ...

Schedule

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 RiemannRoch spaces. 
IV.1  IV.2 

8 
Oct 30 
Weak RiemannRoch theorem. Finiteness of the Picard group of degree 0.
Application to the rationality of the zeta function.

IV.2  IV.3 

9 
Nov 6 
RiemannRoch 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.

VII  VIII 
 
EXAM 
Jan 19  14:00  17:00 in Room 412     
RETAKE 
Feb 21  14:00  17:00 in Room 412     
