Clay Mathematics Institute

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Vol 8. Arithmetic Geometry

Vol 8

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

CMI/AMS publication. 562 pp., Softcover, List: $119, AMS Members: $95.20. Order code: CMIP/8. Students: $95.20. Order code: CLAY MATH. Available at the AMS bookstore

Editors: Henri Darmon, David Ellwood, Brendan Hassett, Yuri Tschinkel