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Arithmetic Geometry
Edited by: Henri Darmon, McGill University, Montreal, Quebec, Canada, David Alexandre Ellwood, Clay Mathematics Institute, Cambridge, MA, Brendan Hassett, Rice University, Houston, TX, and Yuri Tschinkel, Courant Institute of Mathematical Sciences, New York University, NY
A co-publication of the AMS and Clay Mathematics Institute.
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Clay Mathematics Proceedings
2009; 562 pp; softcover
Volume: 8
ISBN-10: 0-8218-4476-8
ISBN-13: 978-0-8218-4476-2
List Price: US$119
Member Price: US$95.20
Order Code: CMIP/8
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See also:

WIN--Women in Numbers: Research Directions in Number Theory - Alina-Carmen Cojocaru, Kristin Lauter, Rachel Pries and Renate Scheidler

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

"This book will interest students doing advanced work in mathematics."

-- SciTech Book News

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