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Course information  

• Teachers:
  Prof. Frits Beukers, (Utrecht),  F.Beukers at math.uu.nl
  Dr. Jan-Hendrik Evertse (Leiden),   evertse at math.leidenuniv.nl
• Examination:
  The examination will consist of four homework assignments and an oral exam. Both the oral exam and the average of the four homework assignments count for 50% in the final grade for the course.
  The course is worth 8 credit points.

  The Study guide gives an overview of what you are supposed to know for the oral exam.

• Time/place:
  Tuesdays February 8-May 24 from 14:00-17:00
  (From February 8-March 22 in Science Park room G3.02, and from March 29-May 24 in room C1.112).
  Directions to Science Park UvA
• Format:
  Each session will consist of two hours lecturing and a one hour exercise class.
• Literature:
  Course notes and homework assignments will be posted on this website.
• Contents (tentative; depending on the time):
  - Algebraic curves (conic sections and rational parametrizations, cubic curves, Bezout's theorem, singularities, genus of a curve).
  - Crash course on algebraic number theory; applications to special cases of the Mordell equation y²=x³+k and the generalized Fermat equation.
  - p-adic numbers, valuations, applications to Diophantine equations.
  - Diophantine approximation, approximation of algebraic numbers by rational numbers, application to Thue equations F(x,y)=m where F is a homogeneous polynomial in two variables, the general Mordell equation and reduction to Thue equations.
  - Three term S-unit equations x₁+ x₂+ x₃=0, where the unknowns are S-units, these are positive or negative integers which are all composed of prime numbers from a prescribed finite set S={ p₁,..., p_t}. Applications to other Diophantine equations.
  - The Thue-Siegel-Roth theorem on the approximation of algebraic numbers by rationals, and its important higher dimensional generalization, the p-adic Subspace Theorem of Schmidt and Schlickewei.
  - Applications of the p-adic Subspace Theorem, in particular to multi-term S-unit equations x₁+...+ x_n=0.

Homework assignments:

• First assignment: Deadline March 8    Solutions
  Points for the exercises: 2,3,3,2
• 2nd assignment: Deadline April 5     Solutions
  Points for the exercises: 1a:5, 1b:5, 1c:5, 1d:5, 2a:5, 2b:5, 3a:3, 3b:6, 3c:5, 3d: 6, 4a:2, 4b: 8. Total: 60. Grade =number of points/6.
• 3rd assignment: Deadline May 10     Solutions
• 4th assignment: Deadline May 27    

Schedule:
(may be subject to changes)

• February 8 (FB)

  Introduction to Diophantine equations, introduction to algebraic curves.
  NOTES ON ALGEBRAIC CURVES
  We shall not treat all of the notes. Just sections 1,3,4 (on singular points), 5,7. Today we covered, besides the introduction, sections 1 and 3 (except Bezout).

  Exercise class: 17,18,19.

• February 15 (FB)
  

  Algebraic curves II
  Today we define and compute singular points of plane projective curves (Section 4), sketch the definition of the genus of a curve (Section 7), rational curves (Section 5), Bezout's theorem (Sections 2 and 4) without proofs.

  Exercise class: 9 (also determine genus, there is a typo in (c) 36xz² must be 36x²z²), 10, 11, 7 (ignore the sentence about non-singular curves), 8a) (hint: set x=1, then x=0).

• February 22 (FB)
  

  Algebraic number theory
  NOTES ON ALGEBRAIC NUMBER THEORY
  Today we briefly recall the basics of algebraic number theory.

  Exercise class: 1.6.1, 1.6.2, 1.6.3(1)(4), 1.6.4, 1.6.5, 1.6.6

• March 1 (FB)
  

  Algebraic number theory and Diophantine equations
  NOTES ON DIOPHANTINE EQUATIONS
  We shall discuss unique factorization in prime ideals in number fields (section 1.3 of the notes). As a first application we consider the solution of Mordell's equation y²+k=x³ in integers x,y.

  Exercise class: 1.1.3, 1.1.4, 1.1.5, 1.1.6.

• March 8 (JHE)
  

  p-adic numbers.
  NOTES ON P-ADIC NUMBERS
  We give an introduction on p-adic numbers. We will certainly not have time to discuss everything that is in the notes. Today, we will go through sections 1-3 or 1-4 (depending on how much time we need).

  Exercise class: 1, 2, 3, 5.

  Deadline of the first homework assignment

• March 15 (JHE)
  

  p-adic analysis and applications to linear recurrence sequences.
  From the notes on p-adic numbers, we skip sections 4 and 6, say a few words on section 5, and treat section 7 (on linear recurrence sequences) in more detail.

  Exercise class: 9,10,11,12.

• March 22 (FB)
  

  Thue equations: application of Skolem's method, application of Diophantine approximation.
  Sections 3.1-3.3 of the notes on Diophantine equations.

  Exercise class: 3.6.1-3.6.3

• March 29 (FB)
  

  Siegel's Lemma and Thue's Diophantine approximation method.
  Sections 3.4,3.5 of the notes on Diophantine equations.

  Exercise class: 3.6.4-3.6.6

• April 5 (FB)
  

  The Siegel-Mahler theorem on S-unit equations.
  NOTES ON THE SIEGEL-MAHLER THEOREM

  Exercise class: 4.4.1, 4.4.4, 4.4.5

  Deadline of the 2nd homework assignment

• April 12 (JHE)
  

  Linear forms in logarithms; effective results on Diophantine equations
  NOTES ON LINEAR FORMS IN LOGARITHMS
  We discuss Section 1 and part of Section 2 of the lecture notes.

  Exercise class: 1,2,4.

• April 19 (JHE)
  

  Effective results on unit equations; the subspace theorem
  NOTES ON THE SUBSPACE THEOREM
  We prove an effective finiteness result on unit equations in two unknowns. Next, we make a start with section 1 of the notes on the subspace theorem.

  Exercise class: 2,3 (of the notes on the subspace theorem).

• April 26
  

  NO COURSE

• May 3 (JHE)
  

  The subspace theorem and the p-adic subspace theorem
  NOTES ON THE P-ADIC SUBSPACE THEOREM
  We continue with the lecture notes on the Subspace Theorem, and discuss part of the material on norm form equations. Next, we start with the lecture notes on the p-adic Subspace Theorem.

  Exercise class: Lecture notes on the Subspace Theorem: Exercises 4,6; Lecture notes on the p-adic Subspace Theorem: Exercises 1,2.

• May 10 (JHE)
  

  The p-adic Subspace Theorem (continued).
  We finish the part on the p-adic Subspace Theorem.

  Exercise class: 3,7,8.

  Deadline of the 3rd homework assignment

• May 17 (FB - last course)
  Overview.

• May 27
  

  Deadline of the 4th homework assignment

  Please send your homework either by ordinary mail to:
  Jan-Hendrik Evertse, Universiteit Leiden, Mathematisch Instituut, Postbus 9512, 2300 RA Leiden
  or a pdf-file by e-mail to evertse@math.leidenuniv.nl
  May 27 is the date that I must have received the homework!

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