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, ...
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.

Web Site

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

Email

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

Course Time

Mondays 9:00am–11:00am

Course Location

Room 401 – 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
 Applications to character sums, to codes, to cryptography, etc.
 Estimates of character sums
 Bounds on the number of rational points
 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:
 Bibliography,
 Notes for the first 3 lectures,
 Notes for the 4th lecture and a first set of exercises,
 Notes for the lecture 5,
 The first Homework assignment: it was to be handed in before the lecture starts on Monday 20th March, preferrably as a typeset document, either in person or by email. There are four exercises in the assignment: you are required to give solutions to only three of them.
 Notes for the 6th lecture,
 The second Homework assignment to be handed in before Tuesday 18th April, preferrably as a typeset document by email. There are three exercises in the assignment.
 Lecture notes for the 7th8th lectures (updated version, now with more examples).
 Notes for the 8th lecture (Chapter on the proof of RH, section IV.0.4 is extra, in case you're interested).
 Lecture notes for 9th lecture.
 Lecture notes for 10th lecture.
 The third Homework assignment, to be handed before Monday 22nd May (last lecture), preferrably as a typeset document. There are three exercises in the assignment.
Note: if you notice typos/mistakes, please send me an email, I'll correct the assignment as soon as possible and upload a new version.
 Notes for 11th lecture (coming soon).
 Notes for 11th and 12th lectures about codes (more examples are to be added).
 Notes for the 2nd part of the last lecture (about character sums).
 The exam.

Exam

The exam will take place on Tuesday, 20th June, from 14:00 to 17:00, in Room 174 of the Snellius. If you have not yet registered for the course, but still wish to sit for the exam, please do so quickly.

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  Feb 6  General introduction, Definition of affine algebraic sets  I.1.1  I.1.2   
2  Feb 13  Affine varieties, dimension, projective space, projective algebraic sets  I.1.3  I.2.3  Read lecture notes, check the proofs, work out the examples. 
3  Feb 20  Projective varieties, relation between affine and projective, dimension and function field of a projective variety  I.2.3  I.2.7  Same 
4  Feb 27  Curves: definition of smoothness, characterizations and consequences  II.1.1  II.1.2  Read lecture notes, check the proofs, work out the examples, and have a go at some exercises. 
5  Mar 6  Rational points, places, divisors. Zeta functions: definition and first properties.  II.2.1  II.2.3  First Homework assignment to be handed in on 20th March (see above). 
 Mar 13  No class     
6  Mar 20  Divisors, divisors of functions, Picard group, RiemannRoch theorem, genus, finiteness of class group.  III.1  III.2  Read lecture notes 
7  Mar 27  More about zeta functions: rationality and functional equation  III.3.1  III.3.3  Second Homework assignment to be handed in before 18th April (see above). 
 Apr 3  No class     
8  Apr 10  Consequences of rationality and functional equation, examples of computations of zeta functions, proof of the Riemann Hypothesis.  III.3.3  IV.0.3   
 Apr 17  No class     
9  Apr 24  Consequences of Riemann Hypothesis, and bounds on the number of rational points: HasseWeil, Serre, Ihara, DrinfeldVladuts, ...  IV.0.5  V.5   
10  May 1  Proof of DrinfeldVladuts bound (via explicit formulas). Application of explicit formulas to equidistribution of zeroes of zeta functions (in some special families of curves)  V.6  V.7   
11  May 8  Errorcorrecting codes: basic definitions, examples. Linear codes: vocabulary and further examples  VI.1  VI.3  Third Homework assignment to be handed in before 22nd May (see above). 
12  May 15  More about codes: bounds on invariants, further examples, codes coming from algebraic geometry, Goppa codes     
13  May 22  Explicit examples of Goppa codes. Application of the Riemann Hypothesis to bounds on character sums. An application to the distribution of squares in finite fields.     
EXAM  June 20th  14:00  17:00 in Room 174     

Examination

 Three homework assignments during the semester (50% of final grade),
 Final exam (50% of final grade).
