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Course information
The Study guide gives an overview of what you are supposed to know for the oral exam.
Schedule:
(may be subject to changes)
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.
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) 36xz^{2} must be 36x^{2}z^{2}), 10, 11, 7 (ignore the sentence about non-singular curves), 8a) (hint: set x=1, then x=0).
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
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 y^{2}+k=x^{3} in integers
x,y.
Exercise class: 1.1.3, 1.1.4, 1.1.5, 1.1.6.
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
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.
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
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
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
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.
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).
NO COURSE
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.
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
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!