# Arithmetic Geometry

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*Henri Darmon; David Alexandre Ellwood; Brendan Hassett; Yuri Tschinkel*

A co-publication of the AMS and Clay Mathematics Institute

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.

Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).

#### Readership

Graduate students and research mathematicians interested in algebraic geometry and number theory.

#### Reviews & Endorsements

This book will interest students doing advanced work in mathematics.

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