Introduction
Definition of NX(p) : the a-ne case
Definition of NX(p) : the scheme setting
How large is NX(p) When p ¿ 8?
More properties of p ¿ NX(p)
The Zeta Point of View
Examples
Examples Where Dim X(C) = 0
Examples Where Dim X(C) = 1
Examples Where Dim X(C) = 2
The Chebotarev Density Theorem for a Number Field
The Prime Number Theorem for a Number Field
Chebotarev Theorem
Frobenian Functions and Frobenian Sets
Examples of S-Frobenian Functions and S-Frobenian Sets
Review of l-adic Cohomology
The l-adic Cohomology Groups
Artin's Comparison Theorem
Finite FIelds : Grothendieck's Theorem
The Case of a Finite Field : The geometric and The Arithmetic Frobenius
The Case of a Finite Field : Deligne's Theorems
Improved Deligne-Weil Bounds
Examples
Variation with p
Auxiliary Results on Group Representations
Characters with Few Values
Density Estimates
The Unitary Trick
The l-adic Properties of NX(p)
NX(p) Viewed as an l-adic Character
Density Properties
About NX(p) - NY (p)
The Archimedean Properties of NX(p)
The Weight Decomposition of the l-adic Character hX
The Weight Decomposition : Wxamples and Applications
The Sato-Tate Conjecture
Equidistribution Statements
The Sato-Tate Correspondence
An l-adic Construction of the Sato-Tate Group
Consequences of the Sato-Tate Conjecture
Examples
Higher Dimension: The Prime Number Theorem and the Chebotarev Density Theorem
The Prime Number Theorem
Densities
The Chebotarev Density Theorem
Proof of the Density Theorem
Relative Schemes
References
Index of Notations
Index of Terms
Biography
Jean-Pierre Serre
"The book is written by a master in the area. It puts the objects it treats into their natural conceptual framework. … The book is highly recommended to anyone interested in the fundamental questions it treats. Those enjoying the mathematics created by Serre will also find pleasure and inspiration in this book."
—Gabor Wiese, Mathematical Reviews, April 2013"This book may be regarded (and can be used) both as an introduction to the modern algebraic geometry, written by one of its creators, and as a research monograph, investigating in depth …"
—B.Z. Moroz, Zentralblatt MATH 1238"the mathematics is exquisite and the presentation is wonderful. … The development of the background mathematics and methodology is crystal clear. … this is another terrific book by Serre: it provides a splendid introduction to both a beautiful arithmetic (-geometric) theme and hugely important mathematical methods pertaining to the given theme. It should tantalize the reader and move him to go into these themes in greater depth, using Serre’s exposition as a high-level road map."
—Michael Berg, MAA Reviews, June 2012
