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# Lectures on N_X(p)

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## Book Description

**Lectures on N _{X}**

**(p)**deals with the question on how N

_{X}(p), the number of solutions of mod p congruences, varies with p when the family (X) of polynomial equations is fixed. While such a general question cannot have a complete answer, it offers a good occasion for reviewing various techniques in l-adic cohomology and group representations, presented in a context that is appealing to specialists in number theory and algebraic geometry.

Along with covering open problems, the text examines the size and congruence properties of N_{X}(p) and describes the ways in which it is computed, by closed formulae and/or using efficient computers.

The first four chapters cover the preliminaries and contain almost no proofs. After an overview of the main theorems on N_{X}(p), the book offers simple, illustrative examples and discusses the Chebotarev density theorem, which is essential in studying frobenian functions and frobenian sets. It also reviews ℓ-adic cohomology.

The author goes on to present results on group representations that are often difficult to find in the literature, such as the technique of computing Haar measures in a compact ℓ-adic group by performing a similar computation in a real compact Lie group. These results are then used to discuss the possible relations between two different families of equations X and Y. The author also describes the Archimedean properties of N_{X}(p), a topic on which much less is known than in the ℓ-adic case. Following a chapter on the Sato-Tate conjecture and its concrete aspects, the book concludes with an account of the prime number theorem and the Chebotarev density theorem in higher dimensions.

## Table of Contents

**Introduction**Definition of

*N*: the a-ne case

_{X}(p)Definition of

*N*: the scheme setting

_{X}(p)How large is

*N*When p → ∞?

_{X}(p)More properties of p ↦

*N*

The Zeta Point of View

_{X}(p)**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) Viewed as an ℓ-adic Character*

**N**

_{X}(p)Density Properties

About

*N*

_{X}(p) - N_{Y}(p*)*

**The Archimedean Properties of**

*The Weight Decomposition of the ℓ-adic Character*

**N**

_{X}(p)*h*

*The Weight Decomposition : Wxamples and Applications*

_{X}**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

Index of Notations

Index of Terms

## Reviews

"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 MATH1238"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