# 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

*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