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Moduli Spaces and Arithmetic Dynamics
Joseph H. Silverman, Brown University, Providence, RI
A co-publication of the AMS and Centre de Recherches Mathématiques.
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CRM Monograph Series
2012; 140 pp; hardcover
Volume: 30
ISBN-10: 0-8218-7582-5
ISBN-13: 978-0-8218-7582-7
List Price: US$59
Member Price: US$47.20
Order Code: CRMM/30
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See also:

Potential Theory and Dynamics on the Berkovich Projective Line - Matthew Baker and Robert Rumely

This monograph studies moduli problems associated to algebraic dynamical systems. It is an expanded version of the notes for a series of lectures delivered at a workshop on Moduli Spaces and the Arithmetic of Dynamical Systems at the Bellairs Research Institute, Barbados, in 2010.

The author's goal is to provide an overview, with enough details and pointers to the existing literature, to give the reader an entry into this exciting area of current research. Topics covered include:

(1) Construction and properties of dynamical moduli spaces for self-maps of projective space.

(2) Dynatomic modular curves for the space of quadratic polynomials.

(3) The theory of canonical heights associated to dynamical systems.

(4) Special loci in dynamical moduli spaces, in particular the locus of post-critically finite maps.

(5) Field of moduli and fields of definition for dynamical systems.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

Readership

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

Reviews

"The monograph is an expanded version of Silverman's lecture notes from a workshop held in May 2010, on the topic of moduli spaces of dynamical systems. It is a pleasure to read, as much as it was a pleasure to attend the lectures. Silverman writes well: he explains the mathematics clearly and includes many explicit examples. Best of all, he has peppered the text with leading questions and side remarks, opening doors to hundreds of potential research projects. It will be a useful reference for students and senior researchers alike. ...Every chapter of this monograph offers multiple entries, many conjectural, in the "Silverman dictionary." For anyone already familiar with both the arithmetic theory of elliptic curves and the traditionally analytic results about these dynamical systems, recent results in arithmetic dynamics make the analogies beautifully apparent; with this book, Silverman helps the rest of us to appreciate the similarities."

-- Laura De Marco, University of Illinois at Chicago

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