Course for third year bachelor and master students -  Fall 2017

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Universiteit Leiden
Mathematisch Instituut

EXAM Exam     Answers to the exercises
Dr. Jan-Hendrik Evertse
office 248,  tel. 071-5277148,   email evertse at math.leidenuniv.nl

Assistant (responsible for grading the homework):
Peter Koymans
office 227a,  tel. 071-5277131,  email p.h.koymans at math.leidenuniv.nl

EC POINTS 6 or 8
The basic course is for 6 EC, but interested students may extend this to 8 EC by studying some additional literature and doing some extra homework. You may decide whether you want to extend your number of EC to 8 after you have obtained your grade for 6EC.

Note: Bachelor students who want to investigate the possibility of letting the EC of this course count towards their master's diploma, are adivsed to contact the chairman of the Exam Committee (Ronald van Luijk, rvl at math.leidenuniv.nl) at their earliest convenience.

EXAMINATION FOR 6EC The examination consists of four homework assignments, issued once in approximately three weeks, and a written exam. The written exam has been scheduled on:

Tuesday January 23, 2018, 14:00-17:00, Snellius room 174.

To get a sufficient final grade of 6 or higher, the grade for the written exam should be at least 5. Then the final grade for the course (for 6EC) is determined by

0.5x(average of the grades for the four homework assignments)+0.5x(grade for written exam),

rounded to the closest half, and rounded up to 6 if it is between 5.5 and 6.

At the written exam questions may be asked about the theorems and proofs of chapters 1-7 of the lecture notes as well as the exercises of the four homework assignments. We will not ask questions about the following sections and parts:

  • Proofs of Theorem 2.10, Lemmas 2.15, 2.16 and Theorem 2.21 (but you have to know the results);
  • Proofs and exercises in Sections 3.1-3.3 (but you have to know the results); you can ignore Section 3.4;
  • Proofs of Theorems 4.8, 4.12, 4.22, 4.24 and 4.25 and Lemmas 4.23, 4.26-4.30;
  • Proof of Theorems 5.12, 5.13; you can ignore Section 5.4;
  • Proof of Theorem 6.9 and the lemmas in the proofs.
EXTENSION TO 8 EC Those of you who want to extend the number of EC-points from 6 to 8, should first get a sufficient grade of 6 or higher for 6EC and then deliver the extra homework assignment, which will be posted below under HOMEWORK. To get the two extra EC, the grade for the extra assignment should be at least 5.5. Then the final grade for 8EC is determined by

0.75x(unrounded grade for 6EC)+0.25x(unrounded grade for extra assignment),

rounded to the closest half, and rounded up to 6 if it is between 5.5 and 6.

You will have to do your written exam before doing this extra assignment.

Chapter 1: Introduction
Chapter 2: Geometry of numbers  (updated September 14)
Chapter 3: Algebraic numbers and algebraic number fields
Chapter 4: Transcendence results (correction Exercise 4.6 November 28)
Chapter 5: Linear forms in logarithms
Chapter 6: Approximation to algebraic numbers by rationals
Chapter 7: The Subspace Theorem

For the two additional EC, you have to study the following two chapters, and submit the extra homework assignment posted below:

Chapter 8: p-adic numbers
Chapter 9: The p-adic Subspace Theorem


Assignments and deadlines:
Assignment 1      Due October 18
Assignment 2      Due November 15
Assignment 3      Due December 6
Assignment 4      Due January 5, 2018
Assignment for two extra EC:
Chapter 8, exercises 8.1,8.2,8.3,8.6,8.7;
Chapter 9, exercises 9.1,9.3,9.5,9.6,9.7.
For 9.2 you get a bonus. Part (b) is more difficult so it is counted more heavily than (a).
Due April 16, 2018
Points for the exercises:
8.1 a)5, b)5; 8.2 5; 8.3 5; 8.6 a)4, b)4, c)4, d)3; 8.7 a)2, b)3, c)5, d)5, e)5
9.1 10; 9.2 a)3, b)7 (maximal 10 points bonus) 9.3 10; 9.5 a)5, b) 5, c)5; 9.6 10; 9.7 a)2, b)3, c)2, d)3
total 110 (120 with bonus); grade = number of points/11

  • We very much prefer that you type your homework in (La)TeX and submit the pdf-file. You may either deliver a hard copy on the beginning of the class on the due date, or send it by email to Peter Koymans, p.h.koymans at math.leidenuniv.nl, on 23:59 of the due date at the latest. If you choose the latter alternative, please write 'DA' in the subject box of your email message so that Peter can easily archive it.
  • Don't forget to put your name and student number (both very well readable if not typed) on your homework!
  • The deadlines are strict! Assignments will be posted on this website three weeks before the deadline, so this shouldn't be a problem if you start to work on the exercises well in time and not on the very last day. In general, if you deliver your homework after the deadline, some points may be subtracted from your grade.
  • In some exercises you will need the so-called house of an algebraic number. You can download here a TeX command and copy-paste it into the pre-ambule of your plain tex file or latex file. Then to get the house of an algebraic number α, say, you simply have to type \house{\alpha}. The command works in plain TeX (the oldest version of TeX) as well as LaTeX.
TIME/PLACE Time:  Wednesdays September 6-December 13, 11:00-12:45, with the exception of October 4, November 8.
Place:  Snellius, room 402

Further information on the schedules can be found on the Course schedule webpage.

PREREQUISITES Algebra 1, Algebra 2 (Groups, rings, fields).
In our course we also need a modest amount of theory of extension of fields and Galois theory (Algebra 3). Knowledge of this is convenient, but not necessary, since we will recall what is needed during the course.
REMARKS This course will not be given in 2018/2019.
This course is recommended for a Master's thesis project in Number Theory.
LITERATURE The following books are not compulsary, but recommended for further reading:
  • A. Baker, Transcendental Number Theory, Cambridge University Press, 1975.
    Gives a broad but very concise introduction to Diophantine approximation. In particular, the book discusses linear forms in logarithms of algebraic numbers.
    ISBN 0-521-20461-5

  • E.B. Burger, R. Tubbs, Making transcendence transparent, Springer Verlag, 2004.
    Gives a relaxed introduction to transcendence theory.
    ISBN 0-387-21444-5

  • J.W.S. Cassels, An Introduction to the Geometry of Numbers, Springer Verlag, 1997 (reprint of the 1971 edition)
    This book gives a broad introduction to the geometry of numbers.
    ISBN 3-540-61788-4

  • W.M. Schmidt, Diophantine Approximation, Springer Verlag, Lecture Notes in Mathematics 785, 1980.
    This book discusses among other things some basics of geometry of numbers, Roth's Theorem on the approximation of algebraic numbers by rational numbers, Schmidt's own Subspace Theorem, and several applications of the latter.
    ISBN 3-540-09762-7

  • C.L. Siegel, Lectures on the Geometry of Numbers, Springer Verlag, 1989.
    This book contains lecture notes of a course of Siegel on the geometry of numbers, given in 1945/46 in New York. The main topics are a proof of Minkowski's 2nd convex body theorem, and a proof of Kronecker's approximation theorem.
    ISBN 3-540-50629-2
  • Mathematics Subject Classification (version MSC2010)
    Official classification of mathematics subjects. The books in the Mathematical Institute library are classified according to this classification. The number theory books are classified under no. 11.

  • Number Theory Web
    Website for the number theory community with many useful links.

  • Online number theory lecture notes
    Long list of downloadable lecture notes on various branches of number theory.

  • MathSciNet, Zentralblatt
    Online mathematical data bases. In principle you can access these databases only by logging on in the university network with your ULCN username and password. I am not sure whether this will work for you (for me it works perfectly), but you may try to access these databases remotely by setting up a proxy connection on your own pc or laptop. For instructions, see Computers at the Mathematical Institute→remote access

  • MacTutor History of Mathematics archive
    An archive with all sorts of facts from the history of mathematics, including biographies of the most important mathematicians.

  • arXiv number theory preprints
    arXiv preprint server with link to the number theory preprints
    (tentative, may be subject to change)
    Diophantine approximation deals with problems such as whether a given number is rational/irrational, algebraic/transcendental and more generally how well a given number can be approximated by rational numbers or algebraic numbers. Techniques from Diophantine approximation have been vastly generalized, and today they have many applications to Diophantine equations, Diophantine inequalities, and Diophantine geometry. Our present plan is to discuss the following topics. But this may be subject to changes.

    Geometry of numbers and applications to Diophantine inequalities. Geometry of numbers is concerned with the study of lattice points (points in ℤn) lying in certain bodies in ℝn. We will discuss the two Minkowski's convex bodies theorems. A set C⊂ℝn is called convex if for any two points in C, the line segment connecting these two points is also in C. A closed symmetric convex body in ℝn is a closed, bounded convex set in ℝn which is symmetric about the origin and has the origin as an interior point. Minkowski's first convex body theorem states that if a closed symmetric convex body C⊂ℝn has volume V(C)≥ 2n, then C contains at least one non-zero lattice point.
    Minkowski's second convex body theorem, which is a generalization of the first, deals with the successive minima of a closed, symmetric convex body C⊂ℝn. For λ>0, let λC denote the body obtained by multiplying all points in C by λ. Then the i-th minimum λi of C is the smallest positive λ such that λC contains i linearly independent lattice points from ℤn. Thus a closed, symmetric convex body C⊂ℝn has n successive minima λ1≤...≤ λn. Now Minkowski's second convex body theorem states that the product of the n successive minima of C is about the inverse of the volume of C, more precisely, (2n/n!)V(C)-1≤λ1... λn≤2n V(C)-1.

    Transcendence. We discuss among others the results of Hermite and Lindemann on the transcendence of e and π.

    Approximation of algebraic numbers by rationals. A well-known theorem (which can be deduced for instance using Dirichlet's box principle or Minkowski's convex body theorem but which was known before) asserts that for every real, irrational number α there are infinitely many pairs of integers (x,y) with y>0 and |α-(x/y)|≤ y-2.
    In 1955, K. Roth proved a famous result, stating that if α is an algebraic number, then the exponent -2 on y in the above theorem cannot be replaced by something smaller, more precisely:

    Let α be a real, irrational algebraic number. Then for every d>0 there are only finitely many pairs of integers (x,y) with y>0 and |α-(x/y)|≤ y-2-d.

    Roth's theorem was a culmination of earlier work by Liouville, Thue, Siegel, Gel'fond and Dyson.
    We also intend to discuss a higher dimensional generalization of Roth's Theorem, the so-called Subspace Theorem by W.M. Schmidt (1972), which deals among other things with the simultaneous approximation of algebraic numbers by rationals. This Subspace Theorem is an extremely powerful tool in Diophantine approximation, with many applications to Diophantine equations, linear recurrence sequences, etc.

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