**Astérisque**

Volume: 334;
2010;
284 pp;
Softcover

MSC: Primary 14; 30; 13; 12;
**Print ISBN: 978-2-85629-294-5
Product Code: AST/334**

List Price: $105.00

AMS Member Price: $86.40

# La Droite de Berkovich Sur \(Z\)

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*Jérôme Poineau*

A publication of the Société Mathématique de France

This text is devoted to the study of the Berkovich line over the ring of integers of a number field. It is a geometric object which naturally contains complex analytic lines (or their quotient by conjugation), associated to the infinite places, and classical Berkovich lines over complete valued fields, associated to the finite places. The authors prove that it satisfies nice properties, both from the topological and algebraic points of view. They also provide a few examples of Stein spaces that are contained in this line. The authors explain how this theory may be used to address various questions about convergent arithmetic power series: prescribing zeroes and poles, proving that global rings are Noetherian or constructing Galois groups over them. Typical examples of such series are given by holomorphic functions on the complex open unit disc whose Taylor developments in \(0\) have integer coefficients.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

#### Readership

Graduate students and research mathematicians interested in the Berkovich line.