SOLVABILITY OF DIOPHANTINE EQUATIONS

Instructional Conference, May 7-11, 2007, Lorentz Center.

Back to the website of the instructional conference  |  Prerequisites  |  Hand-outs and exercises


PROGRAM

The lectures are in Room 201 on the second floor and the social activities in the Common room on the third floor. On Monday-Thursday mornings there are lectures. In the afternoons there are exercise sessions where the participants can practice with the theory discussed in the morning lectures. On Friday there will be a series of lectures meant for a general audience.

MONDAY MAY 7

 
09:30-10:00 Picking up badges, hand-outs etc. from Common Room
10:00-10:50 Samir Siksek: Preliminaries on curves of genus 1
11:00-11:50 Yann Bugeaud: Preliminaries on linear forms in logarithms
14:00-16:00 Exercise session
16:00-17:00 Discussion of solutions
17:00-18:00 Wine and cheese party in Common Room

TUESDAY MAY 8

 
09:00-09:50 Nils Bruin: Preliminaries on Chabauty's method
10:00-10:50 Samir Siksek: Preliminaries on Modular/Frey curves
11:00-11:50 Yann Bugeaud: Linear forms in logarithms II
14:00-16:00 Exercise session
16:00-17:00 Discussion of solutions

WEDNESDAY MAY 9  

 
09:00-09:50 Nils Bruin: Chabauty's method II
10:00-10:50 Samir Siksek: Modular/Frey curves II
11:00-11:50 Mike Bennett:The hypergeometric method
14:00-16:00 Exercise session
16:00-17:00 Discussion of solutions
17:00-22:00 Boat trip on Kager plassen and dinner on boat.
The bus will depart on 17:15.

THURSDAY MAY 10  

 
09:00-09:50 Mike Bennett: Modular/Frey curves III
10:00-10:50 Nils Bruin: Chabauty's method III
11:00-11:50 Yann Bugeaud: Linear forms in p-adic logarithms
14:00-16:00 Exercise session
16:00-17:00 Discussion of solutions

FRIDAY MAY 11  

 
11:00-12:00 Johnny Edwards (Utrecht): x2+ y3= z5 and similar equations.
13:30-14:30 Nils Bruin (SFU, Burnaby): Some hyperbolic cases of the generalized Fermat equation.
14:45-15:45 Frits Beukers (Utrecht): Integral points on cubic surfaces.
16:00-17:00 Rob Tijdeman (Leiden): Arithmetical properties of arithmetical progressions.


PREREQUISITES:

Algebraic number theory:
Ideals, unit groups, S-unit groups, class groups, completions, p-adic valuations, p-adic analysis.

S. Lang, Algebraic Number Theory, Chapters 1,2,7 or
J. Neukirch, Algebraic Number Theory, Chapters 1,2

More about p-adic analysis (basics about p-adic power series):
Z.I Borevich, I.R. Shafarevich, Number Theory, Chapter 4

Algebraic curves and elliptic curves:
Function fields, divisors, Picard group, differentials, basic facts about elliptic curves and reduction modulo p of an elliptic curve (vague idea of how this works). Part of this will be recalled briefly during the course.

J.H. Silverman, The arithmetic of elliptic curves,
Chapters 1,2,
Chapter 3 sections 1,2,3; sections 4,5 will be useful,
Chapter 7, sections 1,2; sections 3,4,5 will be useful.

Recommended for further reading:
p-adic methods and some facts about genus 2 curves (not required but useful)

N.P. Smart, The effective resolution of Diophantine equations, Chapter 2, Chapter 14, sections 1,2
J.W.S. Cassels, E.V. Flyn, Prolegomena to a Middlebrow Arithmetic of Curves of genus 2, Chapters 1,2,13


Last modified by Jan-Hendrik Evertse on .