Lectures on NX(p) deals with the question on how NX(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 NX(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 NX(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 NX(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
Definition of NX(p) : the a-ne case
Definition of NX(p) : the scheme setting
How large is NX(p) When p → ∞?
More properties of p ↦ NX(p)
The Zeta Point of View
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
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
Variation with p
Auxiliary Results on Group Representations
Characters with Few Values
The Unitary Trick
The ℓ-adic Properties of NX(p)
NX(p) Viewed as an ℓ-adic Character
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
The Sato-Tate Correspondence
An ℓ-adic Construction of the Sato-Tate Group
Consequences of the Sato-Tate Conjecture
Higher Dimension: The Prime Number Theorem and the Chebotarev Density Theorem
The Prime Number Theorem
The Chebotarev Density Theorem
Proof of the Density Theorem
Index of Notations
Index of Terms
"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