Mathematisches Institut Georg-August-Universität Göttingen

Clay Mathematics Institute Summer School "Arithmetic geometry"

Week 1

	July 17	July 18	July 19	July 20	July 21
10:00	Y. Tschinkel Introduction	Y. Tschinkel Hypersurfaces	Y. Tschinkel Height zeta functions	Y. Tschinkel Toric varieties	Y. Tschinkel Compactifications of additive groups
11:30	B. Hassett Rational surfaces over algebraically closed fields	B. Hassett Effective cones of rational surfaces	B. Hassett Rational surfaces over non-closed fields	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
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 Integral points	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	P. Charollois Frey's method and ternary Diophantine equations Subsequent BBQ

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 Dynamical systems defined by polynomials	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 Arakelov geometry and equidistribution
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 Grothendieck conjecture
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 density of ordinary Hecke orbits
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 Theory of displays I	W. Messing Theory of displays II	Y. I. Manin Iterated modular symbols I	D. Kaledin Cartier isomorphism I	D. Kaledin Cartier isomorphism II