Mathematisches Institut
Georg-August-Universität
Göttingen



   Clay Mathematics Institute Summer School 2006 on "Arithmetic geometry" Recorded Lessons
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Week 1



July 17
July 18
July 19
July 20
July 21
10:00
Y. Tschinkel
Introduction
Y. Tschinkel
Hypersurfaces
Y. Tschinkel
Toric varieties I
Y. Tschinkel
Toric varieties II
Y. Tschinkel
Flag varieties
11:30
B. Hassett
Rational surfaces over algebraically closed fields
B. Hassett
Rational surfaces over non-closed fields I
B. Hassett
Rational surfaces over non-closed fields II
B. Hassett
Singular Del Pezzo surfaces
B. Hassett
Cox rings and universal torsors
14:30
A. Kresch
Brauer groups, Galois cohomology
A. Kresch
Brauer-Manin obstruction with quaternion algebras
A. Kresch
Descent, torsors
A. Kresch
Hasse principle and Brauer-Manin obstruction
A. Kresch
Further examples
16:00
S. Wiedmann
Introduction to computers at the MI
G. Pfister

J. Jahnel
Rational points on hypersurfaces
M. Stoll
Computations with genus 2 curves
M. Stoll
Computations with genus 2 curves II
17:30
BBQ

Exercises

Exercises

Exercises

Exercises





Week 2



July 24
July 25
July 26
July 27
July 28
10:00
H. Darmon
Arithmetic of curves: overview
H. Darmon
Faltings' theorem I
H. Darmon
Faltings' Theorem II
H. Darmon
Modular curves and Mazur's Theorem
M. Rebolledo
Merel's theorem, continued
11:30
J. Starr
Tsen-Lang theorem
J. Starr
Arithmetic over function fields of curves
J. Starr
Arithmetic over function fields of surfaces
D. Harari
Bielliptic surfaces
D. Harari
Enriques surfaces
14:30
C. Popescu
Special values of L-functions
Y. Tschinkel
Compactifications of additive groups
D. Harari
Nonabelian descent
M. Rebolledo
Merel's theorem
F. Bogomolov
Geometry over small fields
16:00
H. Chapdelaine
The generalised Fermat equation
B. Moroz
Circle method I
S. Wiedmann
Zeta functions of regular graphs
B. Moroz
Circle method II
H. Darmon
Fermat curves and Wiles' Theorem
17:30
Exercises

Exercises

Exercises

Exercises

Exercises





Week 3



July 31
August 1
August 2
August 3
August 4
10:00
H. Darmon
Elliptic curves and modular forms
H. Darmon
The theorems of Gross-Zagier and Kolyvagin
H. Darmon
Proof of Kolyvagin's Theorem
H. Darmon
p-adic uniformisation
H. Darmon
Stark-Heegner points
11:30
D. Abramovich
Geometry and arithmetic of curves
D. Abramovich
Kodaira dimension
D. Abramovich
Campana's program
D. Abramovich
The minimal model program
D. Abramovich
Vojta, Campana and ABC
14:30
A. Chambert-Loir
Equidistribution on the projective line
J. Voight
Some diophantine applications of Heegner points
A. Chambert-Loir
Arakelov geometry and equidistribution
M. Greenberg
Calculating Heegner points via overconvergent modular symbols
A. Chambert-Loir
Equidistribution on Berkovich spaces
16:00
N. Elkies
Newton iteration for simultaneous algebraic equations
O. Labs
Construction of hypersurfaces with singularities
J. Voight
Shimura curve computations
J. Schicho
Construction of rational points on rational surfaces
M. Greenberg
Elliptic curves over imaginary quadratic fields
17:30
Exercises

Exercises

Exercises

Exercises

Exercises





Week 4



August 7
August 8
August 9
August 10
August 11
10:00
F. Oort
Introduction: Density of Hecke orbits
F. Oort
The Tate-conjecture
F. Oort
A conjecture of Manin and the weak Grothendieck conjecture
F. Oort
Purity and deformations of p-divisible groups
F. Oort
Proof of the density of ordinary Hecke orbits
11:30
C.-L. Chai
Serre-Tate theory
C.-L. Chai
Dieudonné and Cartier modules
C.-L. Chai
Hilbert modular varieties
B. Poonen
Varieties over finite fields II
C.-L. Chai
Proof of the Grothendieck conjecture
14:30
E. Ullmo
The André-Oort conjecture and Manin-Mumford
B. Poonen
Varieties over finite fields I
E. Ullmo
Equidistribution of special varieties
Yu. I. Manin
Iterated modular symbols II
Yu. I. Manin
Iterated modular symbols III
16:00
W. Messing (ca. 800 MB)
Theory of displays I
W. Messing (ca. 750 MB)
Theory of displays II
Y. I. Manin
Iterated modular symbols I
D. Kaledin
Cartier isomorphism I
D. Kaledin
Cartier isomorphism II