SummerSchool "Arithmetic geometry" Tschinkel - Introduction | 2006

lecture notes: https://drive.google.com/file/d/1VLucSK53-iLrVUbPAanNZ6Lb7nAAgaQ1/view?usp=sharing
Clay Mathematics Institute Summer School 2006 on "Arithmetic geometry"

survey lectures given at the 2006 Clay Summer School on Arithmetic Geometry at the Mathematics Institute of the University of Göttingen. Intended for graduate students and recent Ph.D.'s, this volume will introduce readers to modern techniques and outstanding conjectures at the interface of number theory and algebraic geometry.

The main focus is rational points on algebraic varieties over non-algebraically closed fields. Do they exist? If not, can this be proven efficiently and algorithmically? When rational points do exist, are they finite in number and can they be found effectively? When there are infinitely many rational points, how are they distributed?

For curves, a cohesive theory addressing these questions has emerged in the last few decades. Highlights include Faltings' finiteness theorem and Wiles's proof of Fermat's Last Theorem. Key techniques are drawn from the theory of elliptic curves, including modular curves and parametrizations, Heegner points, and heights.

The arithmetic of higher-dimensional varieties is equally rich, offering a complex interplay of techniques including Shimura varieties, the minimal model program, moduli spaces of curves and maps, deformation theory, Galois cohomology, harmonic analysis, and automorphic functions. However, many foundational questions about the structure of rational points remain open, and research tends to focus on properties of specific classes of varieties.

Contents
About the cover: Rational points on a K3 surface
Noam Elkies
Curves
Rational points on curves
Henri Darmon
Non-abelian descent and the generalized Fermat equation
Hugo Chapdelaine
Merel's theorem on the boundedness of the torsion of elliptic curves
Marusia Rebolledo
Generalized Fermat equations
Pierre Charollois
Heegner points and Sylvester's conjecture
Samit Dasgupta and John Voight
Shimura curve computations
John Voight
Computing Heegner points arising from Shimura curve parametrizations
Matthew Greenberg
The arithmetic of elliptic curves over imaginary quadratic fields and Stark-Heegner points
Matthew Greenberg
Lectures on modular symbolsLectures on modular symbols
Yuri I. Manin
Surfaces
Rational surfaces over nonclosed fields
Brendan Hassett
Non-abelian descent
David Harari
Mordell-Weil Problem for Cubic Surfaces, Numerical Evidence
Bogdan Vioreanu
Higher-dimensional varieties
Algebraic varieties with many rational points
Yuri Tschinkel
Birational geometry for number theorists
Dan Abramovich
Arithmetic over function fields
Jason Starr
Galois + Equidistribution=Manin-Mumford
Nicolas Ratazzi and Emmanuel Ullmo
The Andre-Oort conjecture for products of modular curves
Emmanuel Ullmo and Andrei Yafaev
Moduli of abelian varieties and p-divisible groups
Ching-Li Chai and Frans Oort
Cartier isomorphism and Hodge Theory in the non-commutative case
Dmitry Kaledin
http://www.uni-math.gwdg.de/aufzeichnungen/SummerSchool/

Видео SummerSchool "Arithmetic geometry" Tschinkel - Introduction | 2006 канала Graduate Mathematics