**Mathematical Surveys and Monographs**

Volume: 222;
2017;
344 pp;
Hardcover

MSC: Primary 11; 14; 53;

Print ISBN: 978-1-4704-3623-0

Product Code: SURV/222

List Price: $124.00

Individual Member Price: $99.20

**Electronic ISBN: 978-1-4704-4089-3
Product Code: SURV/222.E**

List Price: $124.00

Individual Member Price: $99.20

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# Foundations of Arithmetic Differential Geometry

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*Alexandru Buium*

The aim of this book is to introduce and develop an arithmetic
analogue of classical differential geometry. In this new geometry the
ring of integers plays the role of a ring of functions on an infinite
dimensional manifold. The role of coordinate functions on this
manifold is played by the prime numbers. The role of partial
derivatives of functions with respect to the coordinates is played by
the Fermat quotients of integers with respect to the primes. The role
of metrics is played by symmetric matrices with integer
coefficients. The role of connections (respectively curvature)
attached to metrics is played by certain adelic (respectively global)
objects attached to the corresponding matrices.

One of the main conclusions of the theory is that the spectrum of
the integers is “intrinsically curved”; the study of this
curvature is then the main task of the theory. The book follows, and
builds upon, a series of recent research papers. A significant part of
the material has never been published before.

#### Table of Contents

# Table of Contents

## Foundations of Arithmetic Differential Geometry

#### Readership

Graduate students and researchers interested in algebraic geometry, number theory, and algebraic groups.