Lectures on N[x](p)

Introduction 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 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