Lectures on N_X(p)

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A01=Jean-Pierre Serre
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Algebraic Closure
Algebraic Integer
Author_Jean-Pierre Serre
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Auxiliary Results on Group Representations
Brauer Character
Category1=Non-Fiction
Category=PBH
Category=PBMW
Cl ΓS
class
Cohomology Group
Compact Subgroup
conjugacy
Conjugacy Class
COP=United Kingdom
curve
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elliptic
Elliptic Curve
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finite
Finite Field
Finite Galois Extension
Finite Type
Frobenius Endomorphism
galois
Galois Extension
Galois Group
group
haar
Haar Measure
Higher Dimension: The Prime Number Theorem and the Chebotarev Density Theorem
Introduction
Language_English
Maximal Compact Subgroup
Maximal Ideal
measure
Mod Pe
Number Field
Open Normal Subgroup
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Permutation Character
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Prime Number Theorem
PS=Forthcoming
Smooth Projective Variety
softlaunch
The Chebotarev Density Theorem for a Number Field
The Sato-Tate Conjecture
type
Virtual Character
Weight Decomposition

Product details

  • ISBN 9781032929088
  • Weight: 254g
  • Dimensions: 152 x 229mm
  • Publication Date: 14 Oct 2024
  • Publisher: Taylor & Francis Ltd
  • Publication City/Country: GB
  • Product Form: Paperback
  • Language: English
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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.