Mathematisches Institut GeorgAugustUniversität 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 nonclosed fields I 
B. Hassett Rational surfaces over nonclosed 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 BrauerManin obstruction with quaternion algebras 
A. Kresch Descent, torsors 
A. Kresch Hasse principle and BrauerManin 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 TsenLang 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 Lfunctions 
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 GrossZagier and Kolyvagin 
H. Darmon Proof of Kolyvagin's Theorem 
H. Darmon padic uniformisation 
H. Darmon StarkHeegner 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. ChambertLoir Equidistribution on the projective line 
J. Voight Some diophantine applications of Heegner points 
A. ChambertLoir Arakelov geometry and equidistribution 
M. Greenberg Calculating Heegner points via overconvergent modular symbols 
A. ChambertLoir 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 Tateconjecture 
F. Oort A conjecture of Manin and the weak Grothendieck conjecture 
F. Oort Purity and deformations of pdivisible groups 
F. Oort Proof of the density of ordinary Hecke orbits 
11:30 
C.L. Chai SerreTate 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 AndréOort conjecture and ManinMumford 
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 