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Description
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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.
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Web Site
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pub.math.leidenuniv.nl/~griffonrmm/CFF.html
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E-mail
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r.m.m.griffon@math.leidenuniv.nl
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Course Time
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Mondays 9:00am–11:00am
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Course Location
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Room 401 – Snellius Gebouw (Universiteit Leiden)
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Tentative Syllabus
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- 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
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Prerequisites
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Algebra 1,2,3; some basic notions of number theory.
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Lecture Notes
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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 e-mail. 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 e-mail. There are three exercises in the assignment.
- Lecture notes for the 7th-8th 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 e-mail, 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.
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Exam
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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.
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Literature
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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,
- ...
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Schedule
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| 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, Riemann-Roch 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: Hasse-Weil, Serre, Ihara, Drinfeld-Vladuts, ... | IV.0.5 - V.5 | -- |
| 10 | May 1 | Proof of Drinfeld-Vladuts 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 | Error-correcting 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 | -- | -- |
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Examination
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- Three homework assignments during the semester (50% of final grade),
- Final exam (50% of final grade).
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