Lectures on Nx(p)
- Jean-Pierre Serre.
- Boca Raton, FL : CRC Press, c2012.
- Physical description
- ix, 163 p. : ill. ; 24 cm.
- Chapman & Hall/CRC research notes in mathematics series.
Math & Statistics Library
|QA161 .P59 S44 2012||Unknown|
- Serre, Jean-Pierre, 1926-
- Includes bibliographical references (p. 147-155) and indexes.
- 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.
- (source: Nielsen Book Data)9781466501928 20160607
- Publisher's Summary
- 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.
(source: Nielsen Book Data)9781466501928 20160607
- Publication date
- Research notes in mathematics ; v. 11
- On t.p. "x" is subscript.
- "An AK Peters book."
- 9781466501928 (hardback)
- 1466501928 (hardback)
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