Vol 8. Arithmetic Geometry

This book is based on 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