Lectures on N_X(p)

This book presents several basic techniques in algebraic geometry, group representations, number theory, â€žâ€œ-adic and standard cohomology, and modular forms. It explores how NX(p) varies with pwhen the family (X) of polynomial equations is fixed. The text examines the size and congruence properties ofNX(p)and describes the ways in which it is computed. Along with covering open problems and offering simple, illustrative examples, the author presents various theorems, including the Chebotarev density theorem and the prime number theorem.

Preface	p. vii
Conventions	p. ix
Introduction	p. 1
Definition of N_x(p): the affine case	p. 1
Definition of N_x(p): the scheme setting	p. 1
How large is N_x(p) when p → ∞ ?	p. 2
More properties of p → N_x(p)	p. 4
The zeta point of view	p. 5
Examples	p. 7
Examples where dim X(C) = 0	p. 7
Examples where dim X(C) = 1	p. 10
Examples where dim X(C) = 2	p. 12
The Chebotarev density theorem for a number field	p. 15
The prime number theorem for a number field	p. 15
Chebotarev theorem	p. 17
Frobenian functions and frobenian sets	p. 20
Examples of S-frobenian functions and S-frobenian sets	p. 25
Review of l-adic cohomology	p. 31
The l-adic cohomology groups	p. 31
Artin's comparison theorem	p. 32
Finite fields: Grothendieck's theorem	p. 33
The case of a finite field: the geometric and the arithmetic Frobenius	p. 34
The case of a finite field: Deligne's theorems	p. 35
Improved Deligne-Weil bounds	p. 36
Examples	p. 40
Variation with p	p. 42
Auxiliary results on group representations	p. 45
Characters with few values	p. 45
Density estimates	p. 56
The unitary trick	p. 59
The l-adic properties of N_x(p)	p. 65
N_x (p) viewed as an l-adic character	p. 65
Density properties	p. 73
About N_x(p) - NY(p)	p. 78
The Archimedean properties of N_x(p)	p. 83
The weight decomposition of the l-adic character h_x	p. 83
The weight decomposition : examples and applications	p. 90
The Sato-Tate conjecture	p. 101
Equidistribution statements	p. 101
The Sato-Tate correspondence	p. 106
An l-adic construction of the Sato-Tate group	p. 111
Consequences of the Sato-Tate conjecture	p. 114
Examples	p. 121
Higher dimension: the prime number theorem and the Chebotarev density theorem	p. 131
The prime number theorem	p. 131
Densities	p. 134
The Chebotarev density theorem	p. 136
Proof of the density theorem	p. 138
Relative schemes	p. 144
References	p. 147
Index of Notations	p. 157
Index of Terms	p. 161
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